Theoretical Study of Effective Parameters in the Friction Reduction by Ultrasonic Vibrations in Solid Surfaces


 Ultrasonic vibrations are used in many fields to reduce friction forces. In this paper, the reasons for the reduction of friction in solid surfaces are investigated by using the friction model of Dahl and the elastic-plastic contact model. Based on the theoretical model, four parameters: relative velocity, contact surface, distance between surfaces, and Young's modulus, are effective in the frictional force reduction. This study is validated using experimental tests. The results showed that the effects of oscillations of the relative velocity and changes of contact surfaces on the friction reduction are 51% and 12%, respectively. The minimum effect, among the factors, was related to Young's modulus with a value of 1%. The reason for the force reduction is the nonlinear behavior of the contact surface, contact stiffness, and the friction force functions. Fluctuations in their input parameters causes an asymmetric oscillation in the value of those functions. This feature changes the average value of these functions and therefore reduces the friction forces.


Introduction
Today, the use of ultrasonic waves in different industries has been extensively progressed. The applications of the ultrasonic waves include medicine, cleaning, welding of metals and plastics, wastewater purification, food industry, ultrasonic sewing, cutting process, stress-relief of industrial components, forming processes, fluid spray in the production of semi-conductors, homogenizer and so on.
An important application of ultrasonic vibrations is the friction force reduction. When two objects slip on a surface, the asperities of both surfaces are engaged, and friction is produced. The sliding friction force between two objects is reduced as the ultrasonic vibrations are applied. This property is significant in machining and forming processes. For example, ultrasonic vibrations are used to reduce the friction of chips and cutting tools, which decreases the force and temperature of cutting tools; besides, the stability of machining processes and the quality of machining surface are increased [1]. Zhu et al. [2] investigated the effect of ultrasonic vibrations on the plasticity of metals during the compression test. The titanium and aluminum metals are used for the tests. They observed that the amplitude and frequency of vibrations affect the plasticity of the metals; however, the lower frequency has a higher effect of "acoustic softening". If contacting surfaces are placed under a microscope, it can be seen that the surfaces have some asperities, and friction is created as these asperities are engaged. When the asperities are engaged, they undergo a deformation that can be elastic, elastic-plastic, or fully plastic. The first attempt in the field of roughness deformation was made by Greenwood et al. [3]. They modeled the contact between a rough surface and a solid surface without asperities, and an elastic model was presented for defining the deformation of asperities. Chang et al. [4] developed the Greenwood model and presented an elastic-plastic model for the deformation of asperities. Polycarpou et al. [5] presented an elastic-plastic model for deformation of asperities which was more accurate than other models.
A theoretical and experimental study was performed by Littmann et al. to investigate the effect of longitudinal and transverse ultrasonic vibrations of the frictional force [6]. They obtained a relation between the relative velocity of two objects and the friction coefficient. Leus and Gutowski [7] examined the Coulomb and Dahl models for Littmann tests. They observed that the results of Dahl model have a better agreement than the Coulomb model. Adachi et al. [8] investigated the friction micro-mechanism in the presence of ultrasonic waves. In their tests, oscillation motions, both circular and linear with respect to a rotating disc, were considered, and they concluded that with increasing the rotational speed of the disc, the friction between surfaces would increase. Besides, by increasing the oscillation amplitude, the friction force is decreased. They reported that in oscillations with low amplitude, the disc's rotational speed has no significant effect on the friction force. In 2014, Dang and Dapino [9] had experimentally and theoretically studied the effect of double-way ultrasonic vibrations (longitudinal and transverse directions) on the friction force between two objects. They obtained the friction force between an aluminum and steel objects under a determined vibration amplitude and normal force. The results showed that the friction force in the presence of two-way ultrasonic vibrations has reduced up to 60%.
In this paper, Dahl's friction model along with the cubic elastic-plastic contact model were used to estimate the friction force in the presence of ultrasonic vibrations. The effects of oscillation terms on the friction force, contact stiffness, and contact surface functions have been examined. Then the reasons for friction force reduction were studied mathematically. Besides, the effect of reduction of Young's module has been included in the friction model. Finally, the effect of each of these parameters on the friction force reduction is investigated.

Theoretical model
When two objects are in contact and slip/roll on each other, a resistant force in the opposite direction, called friction, is created. Different phenomena are observed in friction between two surfaces, each in turn affects the complexity of frictional models that can take them into account.
Dahl model is one of these frictional models. Dahl described the behavior of the friction phenomenon by using the strain-stress diagram. For two bodies slip on each other with small displacement, he observed that the body returns to its initial position. Dahl compares this behavior with the materials property in the elastic region. With further displacements of two surfaces, plastic deformation occurs. The maximum force obtained in the force-displacement diagram is called adhesion force. For soft materials, the pre-rupture force decreases as the displacement increases [10]. The friction force using the Dahl model is defined as follows: where δ is displacement. It is obtained from the following differential equation: where denotes the relative velocity between two objects and is the tangential contact stiffness, which is obtained from contact models. The tangential contact stiffness can be calculated by the cubic model as follows [9]: where , and represent the contact surface of all asperities, and surfaces distance, respectively.
Besides, is Young's module calculated by Eq. (4): where 1 and 2 are the Young's module of two surfaces and 1 and 2 are their corresponded poison ratio, respectively.
When two surfaces are in contact, two main planes can be defined. One is the average height of asperities, and the other one is the average height of the plane on which asperities exist. Fig. 1 shows the specifications of these two planes. In this figure, the upper plane (base plane) is considered to be solid and without any asperities.  [4] According to Fig. 1, z is the asperities height, and d is the distance of surfaces (which measures from the base plane to the average height of asperities). Besides, R denotes the radius of asperities, and h is the distance of surfaces, defined from the base plane to the average height of the plane.
The total number of asperities is equal to: where is the nominal contact surface and is the density of asperities, which is measured experimentally. The total number of asperities on the contact surface is defined as follows: where ( ) is the distribution function of asperities height and is considered to be a Gaussian distribution. The limits of integral, ( , ∞) are considered from the positive interference of asperities on the surface of base plane. The interference is expressed by: According to Hertz theory of contact, the area between two asperities is calculated by Eq. (8) [3]: The vertical force between two asperities is The maximum contact pressure between two asperities is defined by Eq. (10).
From equation 10, can be expressed as follows: Kohn et al. [11] defined the maximum contact pressure, at the beginning of plastic deformation as follows: By substituting Eq. (12) in Eq. (11), the critical interference is obtained from Eq. (13).
where , , , and are the Poison ratio of softer material, the radius of the asperity, yield strength of softer material, and combined Young's module of two objects, respectively. It should be noted that is a function of which is obtained using the following equation: Chang et al. [4] have developed the Greenwood relation and obtained some relations for calculating plastic area and the force between two asperities. The area affected by plastic deformation between two asperities is defined as follows: The contact load of asperities is equal to: Polycarpou et al. [5] presented a model for the elastic-plastic deformation of asperities, which was more accurate than the previous models. They used a modified exponential function for asperities For elastic-plastic condition, using Eqs. (15) and (16), the contact surface area calculated as follows: where is the standard deviation of height of asperities. By solving Eq. (18), the real contact area is obtained from the elastic deformation of asperities. where = and is the nominal contact area. By solving Eq. (19), the real contact area is obtained from plastic deformation of asperities using the following equation: The real contact area, ( ), is calculated using Eq. (22).
Finally, the friction force is obtained as follows: where, is the Coulomb friction force obtained from experiments.

Examining Parameters
Given that many researches have experimentally studied the friction reduction phenomenon in the  The oscillation frequency of 20 kHz is used so that the comparison between the results of the present study with those of reference [9] can be made. According to the literature survey, the maximum reduction in Young's module by applying ultrasonic oscillations is 30%. This reduction is considered for the part connected to the transducer (first body). The value of normal oscillation amplitude (d) was reported to be 1.1 .

Velocity
If the velocity between two bodies increases, the relative displacement of two bodies increases.
Increased displacement in the Dahl model means the increase of strain; besides, in the forcedisplacement diagram, this means an increase of force between two bodies. However, based on the Dahl model, the increase in force cannot exceed the Coulomb force due to separations of asperities.
The relative velocity between two bodies is defined as follows: where 1 is the velocity amplitude of the first body, V 2 is the constant velocity of the second body, and is the vibration frequency. By substituting Eq. (24) in Eq. (23), the following equation is derived for friction force.
where the value of Coulomb force is 55N and the initial value for is 14 . Figs. 2 and 3 show the friction force in terms of the velocity ( 2 ) for 0.005m/s and 0.5m/s based on the Dahl model, respectively. The figures reveal that with increasing the relative velocity, the friction force reaches its maximum value faster.  It can be seen from Fig. 4 that by decreasing the constant velocity component, the effect of oscillations is increased. In this case, the average friction force is decreased.

Contact surface
By applying the longitudinal ultrasonic vibrations, the transversed vibrations are applied to the surface due to the Poison effect. Therefore, the height of asperities will increase/decrease, as a result, the contact surface increases/decreases. In this section, the effect of friction force reduction by applying the transversed vibrations of 1.1 , assuming constant Young's module and velocity, is investigated. Firstly, the effect of parameter on the effective contact surface of two bodies is studied. Fig. 8 shows that with increasing the parameter , the contact surface decreases exponentially. Eq. (22) has been used to plot this curve. The function in terms of exhibit a nonlinear behavior. Thus, the variations of parameter lead to an asymmetrical oscillation around the equilibrium point, which reduces its average value.
This reduction trend is depicted in Fig. 9. Then, the effect of on the contact stiffness is investigated. As shown in Fig. 10, this parameter significantly affects the contact stiffness, a part of which is due to the reduction in the contact area. The function has a nonlinear behavior in terms of d parameter. Thus, the variations of this parameter lead to an asymmetric oscillation around the equilibrium point, as a result, its mean value is reduced. This reduction is represented in Fig. 11.

Young's module
The acoustic softening is occurred in metals, such as steel and aluminum, by applying the ultrasonic vibrations. This phenomenon leads to a reduction of Young's module [4, 13, and 14].    The value of the friction force before applying the ultrasonic vibrations was the same, 55N, both in the experimental test and the theoretical analysis. By considering all parameters, a 62% reduction was observed in the friction force (reaching 21N). With reducing the transversed vibrations to 0.9 , and the velocity oscillations amplitude to115 × 10 4 , the value of friction force was 28N, representing great agreement with the experimental value. The difference in the vibrations' amplitude may be due to the measurement error reported in reference [9].

Conclusion
The elastic-plastic contact and Dahl friction models have been used, and it was concluded that relative velocity, contact surface, distance of surfaces, and Young's modulus affect the friction force reduction under ultrasonic vibrations. The velocity oscillations have the maximum effect.
Besides, the contact surface and reduction of Young's module are respectively the second and third parameters with maximum effect. The experimental tests were used to validate the analytical model. In this research, ultrasonic vibrations caused a 62% reduction in friction force. Based on the modeling results, the oscillation velocity and back-and-forth motion on the force-displacement curve are the most influential parameters in friction reduction. The effect of velocity oscillations is approximately 51%. The results revealed that the transversed vibrations with an amplitude of 1.1 caused a reduction of 4% in the contact surface. In addition, variations of the contact surface and surface distance caused a 12% reduction in friction force. Also contribution of the contact stiffness and young modulus were 11% and 1% respectively.
Mathematically, the nonlinear behaviors of friction force, contact surface and contact stiffness functions lead to an asymmetric oscillation in the value of those functions. This property leads to a reduction in the average value of this function when the oscillations are applied.

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Authors Contributions
Mohammad reza karafi is supervisor of the research. He interprets the models and revises the manuscript. Ali fattahi conducted experiments and performed calculations and prepared the manuscript.

Funding
The authors received no financial support for the research.

Competing Interests
We declare that we have no significant competing financial, professional, or personal interests that might have influenced the performance or presentation of the work described in this manuscript.

Availability of data and materials
Raw data were generated at Metrology and Advanced Mechatronics Laboratory, faculty of mechanical engineering, Tarbiat Modares University, Tehran, IRAN. Derived data supporting the findings of this study are available from the corresponding author on request.

Code availability
Matlab codes used in this researches are available from the corresponding author on request.