Calculation Model of the Effect of Periodic Change of Normal Force on Sliding Friction Characteristics between the Planes

Abstract

How to reduce the friction resistance between two planes is a difficult problem that has been studied in the field of tribology. Aiming at this problem, the concept of reducing the friction resistance by exciting the periodic change of normal force between the planes is proposed. A calculation model of the displacement amplitude distribution of the plate is established based on the principle of reciprocity of work, and the influence of the periodic change of normal force on sliding friction between the planes is studied. Additionally, an experimental device for analysis of friction between the planes under the periodic change of normal force is established. The calculation model of the plate’s displacement amplitude distribution considering the change frequency of normal force is verified and modified by experiments. The research results mainly show two aspects. On the one hand, the calculation model of the displacement amplitude distribution of the plate is in good agreement with the experimental results, which can effectively help to study the effect of periodic change of normal force on sliding friction between the planes. On the other hand, the change of amplitude and frequency of the normal force have an influence on the sliding friction between the planes. That is, the latter decreases with the increase of the former. The above conclusions have great reference significance for the study of vibration drag reduction in engineering production.

1. Model of Influence of Normal Force’s Periodic Change on Sliding Friction Characteristics between the Planes

Reducing the friction resistance of moving surfaces is an important problem in the field of tribology [1,2,3]. In order to solve this problem, a series of engineering methods have been formed, which are as follows: bionic drag reduction [4], chemical drag reduction [5], vibration drag reduction [6], paint drag reduction [7], etc. Research and practice show that vibration drag reduction is one of the most effective methods. To this end, various engineering processes for vibration drag reduction have been formed, such as vibration pile pulling [8], vibration conveying [9], and vibration screening [10]. However, as there are many forms of vibration, the matching relationship between different forms of vibration drag reduction and drag reduction’s working conditions has not been clear, and the research on the influence of different vibration forms on characteristics of friction and drag reduction has not been in-depth. Jae Hyeok Choi et al. studied the influence of contact vibration on nano friction among graphene, Pt, Au and SiOx from the micro mechanism [11], but for the actual engineering requirements, it is also necessary to study the influence of friction between the two Q235A plates on vibration drag reduction under macro conditions. For example, the vibration drag reduction between the drill string and the inner wall of the casing during sliding guidance of oil and gas wells is not enough from the micro perspective, but needs to be studied from the macro perspective. Therefore, in view of the problem of how to reduce friction and drag between two planes, the author proposes the concept of stimulating the periodic change of normal force between the planes to change the friction resistance, establishes models and carries out related experiments so as to provide theoretical support for using normal vibration to reduce drag in engineering.

1.1. Establishment of Calculation Model for the Influence of Normal Force’s Periodic Change on Sliding Friction between the Planes

The study shows [12,13,14] that when the normal force of two plates in contact with each other changes periodically, the high frequency mechanical collision will occur, resulting in the constant change of real contact area. As shown in Figure 1a, when the plate is not subjected to periodic change of normal force, no deformation occurs on the plate. At this time, if there is relative movement between contact surfaces, the friction resistance on the plate is the product of the positive pressure on the upper plate and the friction coefficient.

$$f_1=\mu N_1$$

(1)

Figure 1b shows the deformation of the plate under the action of the changing normal force. When the plate is subjected to the normal force whose changing amplitude is less than the plate’s gravity, the displacement amplitude of the plate in a certain area, near where the normal force is applied, will be generated. In this area the two plates do not contact and are still contact with each other in the rest of areas, which means that the contact area of the two plates decreases compared with that before the deformation. When the plate is subjected to the periodically changing normal force, the displacement amplitude is in a dynamic change motion. Besides, the upper plate with displacement amplitude is in a “weightless state”, that is, when the plate with displacement amplitude is deformed, the positive pressure on the plate and the average friction resistance in relative motion all decrease. At this time, the friction resistance between two plates is:

$$f_2=\mu N_2$$

(2)

1.2. Model Building and Solving

According to the above analysis, as long as the distribution of the displacement amplitude on the contact surface of the plate is obtained, the change of the friction resistance under the excitation state can be got so as to analyze the influence of vibration on the friction resistance.

In many engineering problems, one of the two objects in friction is elastic, such as the stratum in the vibration pile pulling process mentioned above. Therefore, this kind of research problem can be transformed into the force bending model of four-sided free plate on elastic foundation. As shown in Figure 2, a flat plate with free four sides on an elastic foundation is subjected to an arbitrary load (P) on the position (ζ, η), and the displacement of the plate edge is assumed to be:

$$W(x,0)=D_1+(D_2+D_1)x/a+{\displaystyle \sum _{m=1}^{\infty }{B}_1{}_m\sin k_{m}x}$$

(3)

$$W(x,b)=D_3+(D_4+D_3)x/a+{\displaystyle \sum _{m=1}^{\infty }{B}_2{}_m\sin k_{m}x}$$

(4)

$$W(0,y)=D_1+(D_1+D_3)y/b+{\displaystyle \sum _{m=1}^{\infty }{B}_1{}_n\sin k_{m}y}$$

(5)

$$W(a,y)=D_2+(D_4+D_2)y/b+{\displaystyle \sum _{m=1}^{\infty }{B}_2{}_n\sin k_{m}y}$$

(6)

In the formula, $D_1$, $D_2$, $D_3$, $D_4$ represents the displacement at the plate angle ($(0,0)$, $(a,0)$, $(0,b)$, $(a,b)$) respectively, and their units are mm.

At present, numerical method and analytical method are the main methods to deal with plate bending. Numerical calculation methods mainly include finite element method and boundary method [15] which lack universality. While in the analytical method, the displacement function needs to be determined first, and then the unknown parameters need to be calculated according to the boundary conditions, whose process is complicated and is not conducive to engineering application. The use of the reciprocity theorem of work can simplify the solution process [16], and the results have high reliability and strong applicability. The theorem of reciprocity of work means that when two sets of generalized forces act on a linear elastic body, the work done by the first set of force on the displacement caused by the second set of force is equal to the work done by the second set of force on the displacement caused by the first set of force.

Figure 3 shows a four-sided simply supported plate on an elastic foundation. According to the theorem of reciprocity of work, the work done on the displacement caused by the periodically changing normal force in the system of Figure 2 is equal to the work done by the displacement caused by the periodically changing normal force in the system of Figure 3. According to the theory of elastic thin plate, the equilibrium equation of four-sided simply supported plate can be obtained as:

$$\frac{{\partial }^4{w}_1}{\partial x^4}+2\frac{{\partial }^4{w}_1}{\partial x^2\partial y^2}+\frac{{\partial }^4{w}_1}{\partial y^4}+K{w}_1=\frac{\delta (x-\zeta ,y-\eta )}{D}$$

(7)

In the formula, $w_1$ is the vertical displacement of the plate and its unit is mm; $K$ is the stiffness coefficient of the foundation reaction and its unit is N/mm; D is the bending stiffness of the plate and its unit is N/mm; h is the thickness of the plate and its unit is mm; $\delta (x-\zeta ,y-\eta )$ is the Dira Delta function.

By applying the reciprocity principle of work to the systems in Figure 2 and Figure 3, it can be obtained that:

$$\begin{gathered} W(\zeta ,\eta )={\displaystyle \underset{0}{\overset{a}{\int }}[V_{1y}(x,0,\zeta ,\eta )W(x,0)-V_{1y}(x,b,\zeta ,\eta )W(x,b)]}dx+{\displaystyle \underset{0}{\overset{b}{\int }}[V_{1x}(0,y,\zeta ,\eta )W(0,y)-V_{1x}(a,y,\zeta ,\eta )W(a,y)]}dy+ \\ [R_1(a,b,\zeta ,\eta )D_4-R_1(0,b,\zeta ,\eta )D_3]+[R_1(0,0,\zeta ,\eta )D_1-R_1(a,0,\zeta ,\eta )D_2] \end{gathered}$$

(8)

In the formula, $V_{1y}(x,0,\zeta ,\eta )$, $V_{1x}(0,y,\zeta ,\eta )$, $R_1(x,y,\zeta ,\eta )$ are the distributed shear force at the plate edge and plate angular reaction force along the x-axis and y-axis respectively. The boundary condition of the four-sided free vibrating plate is:

$$\frac{{\partial }^{3}W(\zeta ,0)}{\partial {\eta }^3}+(2-v)\frac{{\partial }^{3}W(\zeta ,0)}{\partial \eta \partial {\varsigma }^2}=0$$

(9)

$$\frac{{\partial }^{3}W(\zeta ,b)}{\partial {\eta }^3}+(2-v)\frac{{\partial }^{3}W(\zeta ,b)}{\partial \eta \partial {\varsigma }^2}=0$$

(10)

$$\frac{{\partial }^{3}W(0,\eta )}{\partial {\zeta }^3}+(2-v)\frac{{\partial }^{3}W(0,\eta )}{\partial {\eta }^2\partial \varsigma }=0$$

(11)

$$\frac{{\partial }^{3}W(a,\eta )}{\partial {\zeta }^3}+(2-v)\frac{{\partial }^{3}W(a,\eta )}{\partial {\eta }^2\partial \varsigma }=0$$

(12)

$$\frac{{\partial }^{2}W(0,0)}{\partial \zeta \partial \eta }=0$$

(13)

$$\frac{{\partial }^{2}W(a,0)}{\partial \zeta \partial \eta }=0$$

(14)

$$\frac{{\partial }^{2}W(0,b)}{\partial \zeta \partial \eta }=0$$

(15)

$$\frac{{\partial }^{2}W(a,b)}{\partial \zeta \partial \eta }=0$$

(16)

Substitute (3)–(6) into (8) to get:

$$\begin{gathered} {\displaystyle \sum _{m=1}^{\infty }\frac{B_1{}_n}{2{\lambda }^2}}\left[\frac{\varphi (a_n,n)\sinh a_n(a-\zeta )}{\sinh a_{n}a}-\frac{\varphi ({\beta }_n,n)\sinh {\beta }_n(a-\zeta )}{\sinh {\beta }_{n}a}\right]\sin k_n\eta + \\ {\displaystyle \sum _{m=1}^{\infty }\frac{B_2{}_n}{2{\lambda }^2}}\left[\frac{\varphi (a_n,n)\sinh a_n\zeta }{\sinh a_{n}a}-\frac{\varphi ({\beta }_n,n)\sinh {\beta }_n\zeta }{\sinh {\beta }_{n}b}\right]\sin k_n\eta + \\ {D}_1\left[\frac{(a-\zeta )(b-\eta )}{ab}-{\displaystyle \sum _{m=1}^{\infty }\frac{{\lambda }^2}{m\pi }\left[\frac{\sinh a_m(b-\eta )}{a^2{}_m\sinh a_{m}b}-\frac{\sinh {\beta }_m(b-\eta )}{{\beta }^2{}_m\sinh {\beta }_{m}b}+\frac{2{\lambda }^2(b-\eta )}{a^2{}_m{\beta }^2{}_{m}b}\right]}\sin k_m\zeta \right]+ \\ {D}_2\left[\frac{\zeta (b-\eta )}{ab}-{\displaystyle \sum _{m=1}^{\infty }\frac{{\lambda }^2{(-1)}^m}{m\pi }\left[\frac{\sinh a_m(b-\eta )}{a^2{}_m\sinh a_{m}b}-\frac{\sinh {\beta }_m(b-\eta )}{{\beta }^2{}_m\sinh {\beta }_{m}b}+\frac{2{\lambda }^2(b-\eta )}{a^2{}_m{\beta }^2{}_{m}b}\right]}\sin k_m\zeta \right]+ \\ {D}_3\left[\frac{(a-\zeta )\eta }{ab}-{\displaystyle \sum _{m=1}^{\infty }\frac{{\lambda }^2}{m\pi }\left[\frac{\sinh a_m\eta }{a^2{}_m\sinh a_{m}b}-\frac{\sinh {\beta }_m\eta }{{\beta }^2{}_m\sinh {\beta }_{m}b}+\frac{2{\lambda }^2\eta }{a^2{}_m{\beta }^2{}_{m}b}\right]}\sin k_m\zeta \right]+ \\ {D}_4\left[\frac{\zeta \eta }{ab}-{\displaystyle \sum _{m=1}^{\infty }\frac{{\lambda }^2{(-1)}^m}{m\pi }\left[\frac{\sinh a_m\eta }{a^2{}_m\sinh a_{m}b}-\frac{\sinh {\beta }_m\eta }{{\beta }^2{}_m\sinh {\beta }_{m}b}+\frac{2{\lambda }^2\eta }{a^2{}_m{\beta }^2{}_{m}b}\right]}\sin k_m\zeta \right] \end{gathered}$$

(17)

Eight algebraic equations can be obtained by substituting (17) into (9)–(16) respectively so as to get the unknown parameter ($D_1$, $D_2$, $D_3$, $D_4$, $B_1{}_m$, $B_2{}_m$, $B_1{}_n$, $B_2{}_n$) and then obtain the theoretical solution of the equation. The value of $m,n$ depends on the requirements of calculation accuracy.

1.3. Instance Calculation

In practical engineering, steel and rock are the main materials that generate friction and friction pair. The choice of steel for the plate in the model can make the results more practical. Q235A steel is taken as an example to calculate. When elastic modulus $E=2.0\times {10}^6$ t/m², Poisson’s ratio $v_{}=0.312$, plate size $a=b=0.18$ m, plate thickness h = 0.03 m, the displacement amplitude distribution under load (P) of 5.7N is calculated using the model.

Figure 4 shows the amplitude distribution of plate displacement under the action of periodic normal force.

It can be seen that under the action of periodically changing normal force, different degrees of displacement amplitudes with the action point of the normal force as the center occur on the plate, which decrease along the action point to the four edges. Considering the area of the plate as S, the friction resistance before excitation as f, the area of the displacement amplitude of the plate in the figure is 0.20 S, the friction resistance under the corresponding conditions can be calculated to be 0.80 f according to the relationship between the displacement amplitude of the plate and the friction resistance established in Section 1.1.

2. Establishment and Analysis of an Experimental Device for the Analysis of Friction between Planes under the Action of Periodically Changing Normal Force

In order to verify the reliability of the model, an experimental device for the analysis of friction resistance between planes under the action of periodically changing normal force is established, whose schematic diagram is shown in Figure 5 and Figure 6.

Figure 7 shows the variation curve of Q235A steel and −Q235A steel friction pair’s friction reduction percentage with the amplitude, under different action frequency of normal force. It can be seen that under the same action frequency, the percentage of friction reduction decreases with the increase of the normal force amplitude. Under the action frequency of 40 Hz, when the normal force amplitude is 5.7 N, the percentage of friction reduction is 86.03%; when the normal force amplitude is 22.7 N, the percentage of friction reduction is 26.02% and the friction resistance is reduced by 69.75%. The trend of the curve shows that a higher normal force amplitude is beneficial to improve the drag reduction effect, because a higher normal force amplitude can cause a larger range of displacement amplitudes on the contact surface.

Under the same impact force, the percentage of friction reduction decreases with the increase of the action frequency. When the normal force change amplitude is 5.7 N, the percentage of friction reduction is 94.01%. When the action frequency is 10 Hz, the percentage of friction reduction is 86.07%. When the action frequency is 40 Hz and the friction resistance is reduced by 8.44%. The trend of the curve shows that the drag reduction effect increases with the increase of the normal force’s change frequency, which is because the change of the normal force, which belongs to the simple harmonic force, is generated by the inertial vibration exciter. The measured friction is the average value in a measurement period. When the action frequency increases, the number of actions per unit time also increases, which means that the number of displacement amplitudes of the plate increases so as to be beneficial to the friction reduction.

3. Model Validation and Modification

According to the friction reduction experiment of the periodically changing normal force, it can be seen that the action frequency of the normal force has a certain effect on the friction reduction. However, the displacement amplitude distribution model of the plate does not consider the situation under the action frequency, so the model needs to be modified.

The plate has a displacement amplitude due to the changing normal force, which reduces the friction resistance, so there are:

$$f_0-f_d\times C_f=f_{1(N,f)}·(C_f>0)$$

(18)

In the formula, f₀ represents the friction resistance of the plate without the action of the periodically changing normal force and its unit is N; f_d represents the reduced friction resistance of the plate subjected to periodically changing normal force and its unit is N. C_f represents correction factor of action frequency; f_{1 (N, f)} represents frictional resistance of the plate subjected to periodically changing normal force and its unit is N.

For the convenience of calculation, each item in the formula is expressed as the percentage of friction reduction. That is, all the terms in formula (18) are divided by the friction resistance not affected by the periodically changing normal force.

The frequency correction coefficient is calculated by taking the Q235A steel and -Q235A steel friction pair, impact force of 5.7 N, impact frequency of 40 Hz as example. Under this condition, when f_d/f₀ is 0.80 and f_1(5,40) is 0.86, C_f can be got as 0.175. In the same way, the correction coefficient curves with different frequencies under different amplitude changes of normal force can be obtained, as shown in Figure 8.

According to the frequency correction curves under different normal force’s amplitudes changes in Figure 8, the frequency correction coefficients can be obtained, which are substituted into formula 18 to obtain the percentage of friction reduction after modification.

The comparison between the model results and the experimental results after the modification is shown in Figure 9. The modified model results are more similar to the experimental results, and the error value is reduced. Under this experimental conditions, the average error rate between the calculated results of the modified model and the experimental results is 3.28%, so the model used to explain the friction reduction mechanism of periodically changing normal force is reliable.

4. Conclusions

• The amplitude distribution model of plate displacement is established and the influence of periodically changing normal force on plane sliding friction is studied. According to the model, the change of displacement amplitude of contact surface under the action of periodically changing normal force is the main reason for the change of friction resistance of contact surface. Under the action of the periodically changing normal force, the displacement amplitude of the upper plate deviates from the contact surface and is in the state of “weightlessness in the air”. Then the positive pressure on the contact surface decreases, which is finally manifested as the reduction of friction resistance.
• An experimental device for friction analysis between plates under the action of periodically changing normal force is established to study the influence of normal force’s amplitude change and action frequency on friction. The experiments show that under the action of the periodically changing normal force, the friction decreases with the increase of the action frequency, and decreases with the increase of the normal force’s amplitude change.
• The frequency modification of the model is carried out through the experimental results. The calculation results of modified model are in good agreement with the experimental results, and the error between them is low. Under the experimental conditions, the average error value is 3.28%.
• The model is reliable for calculating the friction resistance between planes under the action of the periodically changing normal force, which can provide a theoretical basis for the technology of using the excited normal force to reduce friction in engineering.