**3. Resonance phenomenon caused by nonlinear frictional behavior**

In this section, the frictional effect of NSB on the dynamic characteristics of the sliding object is discussed. The frequency response of the sliding object is an important consideration for developing a highly accurate feed drive system in machine tool and precision machines using friction compensators.

#### **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:

$$m\ddot{\mathbf{x}} = -F(\mathbf{x}) + P(\mathbf{t})\tag{6}$$

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*0sin*ωt* acting on the mass.

$$m\ddot{\mathbf{x}} = -F(\mathbf{x}) + P\_0 \sin \alpha \mathbf{t} \tag{7}$$

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

#### **Figure 5.**

*The analytical model of the rolling guideway for calculating the dynamic characteristics in the feed direction in consideration of the nonlinear spring behavior of friction.*

internal hysteresis curve aba is calculated with Eq. (2) by replacing the values of *x*<sup>r</sup>

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

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*<sup>s</sup> do not consider. The resonances caused by NSB are depended on *x*<sup>s</sup> [18]. Therefore, the starting rolling displacement and steady-state friction force in the rolling

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

*A x* <sup>þ</sup> *Bx<sup>n</sup>* ð Þ� *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>x</sup>* ð Þ *<sup>x</sup>*≤*xs Fs* � *f* <sup>2</sup>ð Þ *x* ð Þ *x*>*xs*

(3)

bristle model [15] and the generalized Maxwell slip (GMS) model [16].

*The relation between friction force* F *and displacement* x *calculated by Masing rule.*

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and *f*<sup>r</sup> to the displacement and friction at point A.

**Figure 3.**

**Figure 4.**

**218**

region should be introduced into the friction model.

*f x*ð Þ¼

*The proposed virgin loading curve* f *for different shape factor* n*.*

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$$
\beta^2 u'' = -\frac{F(u)}{F\_s} + \chi\_0 \sin \pi \tag{8}
$$

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*<sup>s</sup> 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).

$$G(\beta) = \frac{u\_0}{\gamma\_0} = \frac{(u\_{\text{max}} - u\_{\text{min}})}{2\gamma\_0} \tag{9}$$

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

*The phase plane at frequency of A. (a)* n *= 0.5,* β *= 0.54. (b)* n *= 1.5,* β *= 0.45.*

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

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

dimensionless friction force *F*/*F*<sup>s</sup> at frequencies A and B with *n* = 0.5 and 1.5,

, dimensionless displacement *u*, and

negligible.

**221**

**Figure 7.**

respectively [7].

acceleration *u* , dimensionless velocity *u*<sup>0</sup>

*Nonlinear Frictional Dynamics on Rolling Contact DOI: http://dx.doi.org/10.5772/intechopen.94183*

where *u*<sup>0</sup> 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*<sup>0</sup> = 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

#### **Figure 6.**

*The frequency response functions in the feed direction with the non-dimensional excitation force amplitude* γ*<sup>0</sup> = 0.6 for different shape factor* n*. (a)* n *= 0.5. (b)* n *= 1.5.*

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*β*2

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(FRF) *G* (*β*) is calculated by Eq. (9).

respectively.

harmonic resonance.

**Figure 6.**

**220**

*<sup>u</sup>*<sup>00</sup> ¼ � *F u*ð Þ *Fs*

*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

> *<sup>G</sup>*ð Þ¼ *<sup>β</sup> <sup>u</sup>*<sup>0</sup> *γ*0

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*<sup>s</sup> is a function of shape factor *n*. Ultimately, the motion of the sliding object is described by determining *γ*0,

> <sup>¼</sup> ð Þ *<sup>u</sup>*max � *<sup>u</sup>*min 2*γ*<sup>0</sup>

where *u*<sup>0</sup> 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,

**Figure 6** shows an FRF with dimensionless excitation force amplitude *γ*<sup>0</sup> = 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 reso-

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

*The frequency response functions in the feed direction with the non-dimensional excitation force amplitude*

γ*<sup>0</sup> = 0.6 for different shape factor* n*. (a)* n *= 0.5. (b)* n *= 1.5.*

nance. Frequency B lies between the harmonic resonance and its adjacent superharmonic resonance. Finally, frequency C is higher than the frequency of

þ *γ*<sup>0</sup> *sin τ* (8)

(9)

**Figure 7.** *The phase plane at frequency of A. (a)* n *= 0.5,* β *= 0.54. (b)* n *= 1.5,* β *= 0.45.*

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*<sup>0</sup> , dimensionless displacement *u*, and dimensionless friction force *F*/*F*<sup>s</sup> 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

**Figure 8.** *The phase plane at frequency of B. (a)* n *= 0.5,* β *= 0.90. (b)* n *= 1.5,* β *= 0.71.*

(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 9.**

**Figure 10.**

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*The phase plane at frequency of C. (a)* n *= 0.5,* β *= 1.65. (b)* n *= 1.5,* β *= 1.40.*

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*Time history at the frequency of A. (a)* n *= 0.5,* β *= 0.54. (b)* n *= 1.5,* β *= 0.45.*

**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*<sup>0</sup> [7]. As shown in **Figure 12**, as *γ*<sup>0</sup> increases, the resonant frequency decreases. Furthermore, resonance does not occur even if *γ*<sup>0</sup> 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

*Nonlinear Frictional Dynamics on Rolling Contact DOI: http://dx.doi.org/10.5772/intechopen.94183*

**Figure 9.** *The phase plane at frequency of C. (a)* n *= 0.5,* β *= 1.65. (b)* n *= 1.5,* β *= 1.40.*

**Figure 10.** *Time history at the frequency of A. (a)* n *= 0.5,* β *= 0.54. (b)* n *= 1.5,* β *= 0.45.*

(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

**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*<sup>0</sup> [7]. As shown in **Figure 12**, as *γ*<sup>0</sup> increases, the resonant frequency decreases. Furthermore, resonance does not occur even if *γ*<sup>0</sup> 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 com-

Then, the results of the proposed friction model are compared with the experi-

**Figure 13** shows the relationship between the excitation force amplitude and the

mental results using commercially available roller guideways for verifying 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

pliance at resonance is not high than the case with large *n*.

*The phase plane at frequency of B. (a)* n *= 0.5,* β *= 0.90. (b)* n *= 1.5,* β *= 0.71.*

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are caused by VOAD.

validity.

**222**

**Figure 8.**

model can qualitatively predict the force dependency of the dynamic characteristics

*Relationship between the non-dimensional excitation force amplitude* γ*<sup>0</sup> and the resonance frequency and compliance at the harmonic resonance. (a) Harmonic resonance frequency. (b) Compliance at the harmonic*

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

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

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

**4. Nonlinear behavior of rotational direction induced by friction**

**4.1 Experiment for evaluating the dynamics of linear rolling bearing**

of a rolling bearing in the feed direction.

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**Figure 13.**

*resonance.*

the frequency response function [25].

applied to the free end of the column.

screwed to the shaker.

**225**

**Figure 11.** *Time history at the frequency of B. (a)* n *= 0.5,* β *= 0.90. (b)* n *= 1.5,* β *= 0.71.*

**Figure 12.** *The force amplitude dependency of the frequency response caused by the nonlinear spring behavior. (a)* n *= 0.5. (b)* n *= 1.5.*

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#### **Figure 13.**

**Figure 11.**

**Figure 12.**

*(b)* n *= 1.5.*

**224**

*Time history at the frequency of B. (a)* n *= 0.5,* β *= 0.90. (b)* n *= 1.5,* β *= 0.71.*

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*The force amplitude dependency of the frequency response caused by the nonlinear spring behavior. (a)* n *= 0.5.*

*Relationship between the non-dimensional excitation force amplitude* γ*<sup>0</sup> and the resonance frequency and compliance at the harmonic resonance. (a) Harmonic resonance frequency. (b) Compliance at the harmonic resonance.*

model can qualitatively predict the force dependency of the dynamic characteristics of a rolling bearing in the feed direction.
