**2. Nonlinear friction behavior in microscopic displacement regime**

#### **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

**Figure 1.** *The relationship between the friction force* F *and the displacement* x*.*

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

behavior of contact friction produces resonance and displacement-amplitude

*Tribology in Materials and Manufacturing - Wear, Friction and Lubrication*

The resonance phenomenon of the bearing and the amplitude dependency of the dynamics lead to instability of the positioning control system and the deterioration of positioning accuracy [1–6]. In order to develop more accurate positioning control system and damping element for precision positioning system, it is important to clarify and model the effects of the nonlinear friction behavior on the dynamics of

This chapter describes the nonlinear friction behavior occurred with rolling contact condition and its effect on the dynamics of bearings. Section 2 explains the nonlinear spring behavior (NSB) depending on displacement, as well as the modeling method of the nonlinear frictional behavior. Section 3 describes the influence of nonlinear friction on the dynamics of the bearing with some numerical and experimental results. Section 4 shows the influence of nonlinear friction on the dynamics in directions other than the feed direction of the bearing. Section 5 concludes the chapter and gives the future interest of the research related to rolling friction

**2. Nonlinear friction behavior in microscopic displacement regime**

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

**Figure 1** shows the relationship between the friction force *F* of the sliding object

dependency in the dynamics of bearings.

**2.1 Displacement dependent nonlinearity**

*The relationship between the friction force* F *and the displacement* x*.*

bearings.

behavior.

**Figure 1.**

**216**

**Figure 2.** *The relationship between the sinusoidal excitation force* P *and the displacement* x*.*

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*<sup>0</sup> sin*ωt* acting on the mass where *P*<sup>0</sup> 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*<sup>0</sup> 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*<sup>s</sup> 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:

$$F = \begin{cases} f(\mathbf{x}) & (\mathbf{x} \ge \mathbf{0}) \\ -f(-\mathbf{x}) & (\mathbf{x} < \mathbf{0}) \end{cases} \tag{1}$$

$$F = \begin{cases} F\_r + \mathcal{H}\left(\frac{\mathbf{x} - \mathbf{x}\_r}{\lambda}\right) & \left(\frac{\mathbf{x} - \mathbf{x}\_r}{\lambda} \ge \mathbf{0}\right) \\\\ F\_r - \mathcal{H}\left(-\frac{\mathbf{x} - \mathbf{x}\_r}{\lambda}\right) & \left(\frac{\mathbf{x} - \mathbf{x}\_r}{\lambda} < \mathbf{0}\right) \end{cases} \tag{2}$$

where *f*(*x*) is the virgin loading curve. *F*<sup>r</sup> and *x*<sup>r</sup> 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* 6¼ 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*<sup>r</sup> and *F*<sup>r</sup> 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

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* 6¼ 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

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

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.

By introducing the dimensionless parameters *K*<sup>s</sup> = *F*s/*x*s, *ω*<sup>s</sup>

direction [7]. The equation of motion is as follows:

excitation force *P* (*t*) = *P*0sin*ωt* acting on the mass.

of motion is described as Eq. (8).

*consideration of the nonlinear spring behavior of friction.*

**Figure 5.**

**219**

**3.1 Numerical analysis of dynamics considering nonlinear friction**

In this section, the frictional effect of NSB on the dynamic characteristics of the

**Figure 5** shows an analytical model for calculating the dynamics in the feed

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

*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

*The analytical model of the rolling guideway for calculating the dynamic characteristics in the feed direction in*

<sup>¼</sup> *d f* <sup>2</sup>ð Þ *<sup>x</sup> dx*

 *x*¼*xs*

*d f* <sup>1</sup>ð Þ *x dx*

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

rolling bearing.

 *x*¼*xs*

*f* <sup>1</sup>ð Þ¼ *xs f* <sup>2</sup>ð Þ *xs* (4)

*mx*€ ¼ �*F x*ð Þþ *P t*ð Þ (6)

*mx*€ ¼ �*F x*ð Þþ *P*<sup>0</sup> *sin ωt* (7)

<sup>2</sup> = *K*s/*m*, *u* = *x*/*x*s,

(5)

**Figure 3.**

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

internal hysteresis curve aba is calculated with Eq. (2) by replacing the values of *x*<sup>r</sup> and *f*<sup>r</sup> 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*<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 region should be introduced into the friction model.

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

$$f(\mathbf{x}) = \begin{cases} A(\mathbf{x} + B\mathbf{x}^n) \equiv f\_1(\mathbf{x}) & (\mathbf{x} \le \mathbf{x}\_i) \\ F\_1 \equiv f\_2(\mathbf{x}) & (\mathbf{x} > \mathbf{x}\_i) \end{cases} \tag{3}$$

**Figure 4.** *The proposed virgin loading curve* f *for different shape factor* n*.*

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

$$f\_1(\mathbf{x}\_t) = f\_2(\mathbf{x}\_t) \tag{4}$$

$$\left. \frac{df\_1(\mathbf{x})}{d\mathbf{x}} \right|\_{\mathbf{x} = \mathbf{x}\_\iota} = \frac{df\_2(\mathbf{x})}{d\mathbf{x}} \Big|\_{\mathbf{x} = \mathbf{x}\_\iota} \tag{5}$$

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* 6¼ 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.
