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

#### **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.

#### **Figure 14.**

*Experimental setup for evaluating the vibration characteristics.*

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.

vibration modes are called "feed mode" and "pitching mode" in order from low

*Measured natural vibration modes of causing each resonance peaks in FRF. (a) Feeding mode. (b) Pitching*

**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

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

frequency.

**Figure 16.**

*mode.*

**227**

**Figure 15.**

*Force amplitude dependency of dynamic characteristics.*

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

did not disappear.

### **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

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

**Figure 15.** *Force amplitude dependency of dynamic characteristics.*

#### **Figure 16.**

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

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

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.

**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

displacement of the carriage in the feed direction.

*Experimental setup for evaluating the vibration characteristics.*

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

**Figure 14.**

**226**

100 N in half amplitude and up to 1000 Hz in frequency.

**4.2 Influence of excitation force in rotational direction**

*Measured natural vibration modes of causing each resonance peaks in FRF. (a) Feeding mode. (b) Pitching mode.*

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.

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

nonlubricated condition [28]. Thus, the damping ratio converged to about 0.5%

**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

impulse test with 2000 N in the excitation force was about 0.5% with a

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

The rolling machine element is indispensable for realizing high-precision and high-speed relative motion. In addition, its positioning accuracy is approaching the

However, since the rolling elements and the raceways are mechanically in contact, various non-linear phenomena occur. This complicated phenomenon has to be clear by theoretically and experimentally. In this chapter, the author explained the nonlinear spring behavior of friction that occurs in the rolling contact state and its modeling method. Furthermore, the effect of nonlinear friction on the dynamic characteristics of sliding objects was analyzed numerically. The validity of the model was verified by comparing with the experimental results using the rolling

Finally, it was experimentally shown that the nonlinearity of rolling friction affects the motion in directions other than the feed direction. It is known that rolling friction has velocity dependence and acceleration dependence in addition to

This work was supported by Japan Society for Promotion of Science (JSPS)

the nonlinear spring behavior described in this chapter.

KAKENHI Grant Numbers JP16K21036, JP15J06292.

*f*<sup>1</sup> proposed virgin loading curve in the pre-rolling region *f*<sup>2</sup> proposed virgin loading curve in the rolling region

*F*<sup>r</sup> friction force at the arbitrary motion reversal point

nanometer order, and its importance is expected to increase in the future.

when the excitation force increased further.

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

additional spring due to non-linear friction.

**5. Conclusions**

guide.

**Acknowledgements**

**Nomenclature**

*A*, *B* constants

*F* friction force

**229**

*f* virgin loading curve

*F*<sup>s</sup> steady-state friction force

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

**Figure 17.** *Influence of oil specification on the nonlinearity of natural vibration characteristics in the pitching mode.*

**Figure 18.**

*Influence of excitation force amplitude on the nonlinearity of natural vibration characteristics in the rolling mode. (a) Rolling mode. (b) Dynamic characteristics of rolling mode.*

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

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.

## **5. Conclusions**

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

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

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

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

*Influence of oil specification on the nonlinearity of natural vibration characteristics in the pitching mode.*

*Influence of excitation force amplitude on the nonlinearity of natural vibration characteristics in the rolling*

*mode. (a) Rolling mode. (b) Dynamic characteristics of rolling mode.*

a cradle and moves slightly in the feed direction.

nonlinear spring behavior.

**Figure 17.**

**Figure 18.**

**228**

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 non-linear phenomena occur. This complicated phenomenon has to be clear by theoretically and experimentally. In this chapter, the author explained the nonlinear spring behavior of friction that occurs in the rolling contact state and its modeling method. Furthermore, the effect of nonlinear friction on the dynamic characteristics of sliding objects was analyzed numerically. The validity of the model was verified by comparing with the experimental results using the rolling guide.

Finally, it was experimentally shown that the nonlinearity of rolling friction affects the motion in directions other than the feed direction. It is known that rolling friction has velocity dependence and acceleration dependence in addition to the nonlinear spring behavior described in this chapter.

### **Acknowledgements**

This work was supported by Japan Society for Promotion of Science (JSPS) KAKENHI Grant Numbers JP16K21036, JP15J06292.

## **Nomenclature**




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['] differential operator with respect to nondimensional time *τ*, *d*/*dτ*
