**Abstract**

The rolling machine element is indispensable for realizing high-precision and high-speed relative motion. In addition, its positioning accuracy is approaching the nanometer order, and its importance is expected to increase in the future. However, since the rolling elements and the raceways are mechanically in contact, various nonlinear phenomena occur. This complicated phenomenon must be clear by theoretically and experimentally. This chapter describes the nonlinear friction behavior occurred with rolling contact condition and its effect on the dynamics of bearings. First, the characteristics of the non-linear friction caused by rolling machine elements and the nonlinear friction modeling method using the Masing rule are described. From the numerical analysis using the friction model, it is clarified that the motion accuracy decreases due to sudden velocity variation caused by nonlinear friction. Also, the author show that the resonance phenomenon and force dependency of the dynamic characteristics of rolling machine element due to the nonlinear friction. Finally, the author indicates nonlinear friction influences on the dynamic characteristics in the directions other than the feed direction.

**Keywords:** nonlinear dynamics, friction, rolling machine element, elastic contact, bearing

## **1. Introduction**

Rolling machine elements such as linear rolling bearings and rotational bearings are used in various industrial fields because they realize high precision relative motion at high speed. In addition, since the fluctuation of friction force is small, self-excited vibration called stick-slip is less likely to occur. Though rolling elements such as balls and rollers are always in point or line contact with the raceway, resulting in high surface pressure. The rolling contact condition complicates tribological phenomena such as friction and wear, and dynamic behavior of rolling machine elements.

Linear rolling bearings are widely used in semiconductor manufacturing equipment, ultra-precision positioning systems for machine tool, robots, and other equipment. Conventionally, it was necessary to use non-contact bearings such as hydrostatic linear guideways to achieve positioning with nanometer accuracy. However, it has become possible to achieve nanometer positioning accuracy even in rolling bearings by optimizing the rolling contact condition.

On the other hand, since rolling elements and raceways are in rolling contact with each other, the damping capacity of bearing is much lower than other guideways such as hydrostatic guides and sliding guides. In addition, the non-linear

behavior of contact friction produces resonance and displacement-amplitude dependency in the dynamics of bearings.

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 bearings.

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 behavior.

region. Furthermore, the boundary displacement between the pre-rolling region

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

Now, the sinusoidal excitation force *P*(*t*) = *P*<sup>0</sup> sin*ωt* acting on the mass where *P*<sup>0</sup>

The friction model proposed based on the Masing rule can describe the NSB and

*f x*ð Þ ð Þ *x*≥ 0 �*f*ð Þ �*x* ð Þ *x*< 0

� � *x* � *xr*

*<sup>λ</sup>* <sup>≥</sup> <sup>0</sup> � �

*<sup>λ</sup>* <sup>&</sup>lt; <sup>0</sup> � � (1)

(2)

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

*F* ¼

(

*Fr* <sup>þ</sup> *<sup>λ</sup><sup>f</sup> <sup>x</sup>* � *xr*

*Fr* � *<sup>λ</sup><sup>f</sup>* � *<sup>x</sup>* � *xr*

*λ*

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

*λ* � � *x* � *xr*

**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

and the rolling region is called the starting rolling displacement.

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

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

reversal point [12].

**Figure 2.**

model, *λ* = 2.

**217**

**2.2 Nonlinear friction modeling**

[14]. The Masing rule is formulated as follows:

*F* ¼

8 ><

>:
