J. Cent. South Univ. (2016) 23: 562-569
DOI: 10.1007/s11771-016-3103-3
Comprehensive modeling approach of axial ultrasonic vibration grinding force
HE Yu-hui(何玉辉) 1,2, ZHOU Qun(周群) 1,2, ZHOU Jian-jie(周剑杰) 1,2, LANG Xian-jun(郎献军) 1,2
1. College of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China;
2. State Key Laboratory of High Performance Complex Manufacturing, Changsha 410083, China
Central South University Press and Springer-Verlag Berlin Heidelberg 2016
Abstract: The theoretical model of axial ultrasonic vibration grinding force is built on the basis of a mathematical model of cutting deforming force deduced from the assumptions of thickness of the undeformed debris under Rayleigh distribution and a mathematical model of friction based on the theoretical analysis of relative sliding velocity of abrasive and workpiece. Then, the coefficients of the ultrasonic vibration grinding force model are calculated through analysis of nonlinear regression of the theoretical model by using MATLAB, and the law of influence of grinding depth, workpiece speed, frequency and amplitude of the mill on the grinding force is summarized after applying the model to analyze the ultrasonic grinding force. The result of the above-mentioned law shows that the grinding force decreases as frequency and amplitude increase, while increases as grinding depth and workpiece speed increase; the maximum relative error of prediction and experimental values of the normal grinding force is 11.47% and its average relative error is 5.41%; the maximum relative error of the tangential grinding force is 10.14% and its average relative error is 4.29%. the result of employing regression equation to predict ultrasonic grinding force approximates to the experimental data, therefore the accuracy and reliability of the model is verified.
Key words: cutting deformation force; ultrasonic vibration assisted grinding (UVAG); regression equation; comprehensive modeling
1 Introduction
Axial ultrasonic-vibration-assisted grinding (AUVAG) is a machining technology, which exerts the ultrasonic vibration on the workpiece or wheel along the axis direction of the wheel and is referred to as ultrasonic grinding in this work. Features like ultrasonic shock, reciprocating ironing pressure, ultrasound lubrication, energy focused, fast cutting, during the grinding process are brought into full play due to the introduction of ultrasonic vibrations to significantly reduce the grinding force and improve surface integrity of workpiece and stability of the cutting system [1-4]. During the grinding process, grinding force is an important indicator to evaluate the process of grinding, and its size exerts a direct influence on lifecycle of the wheel, temperature of grinding zone, quality workpiece surface, grinding specific energy and so on. Therefore, it is of vital importance to build a model for grinding force prediction in order to scientifically, accurately and effectively control the values of grinding force during the grinding process.
Great progress has been made in modeling studies on the grinding force in terms of ultrasonic grinding. QIN et al [5] established a physics-based predictive cutting force model in ultrasonic vibration assisted grinding for titanium. and SHIMADA et al [6] used statistics to derive the model of ultrasonic grinding force, and conducted an ultrasonic vibration assisted grinding experiment on the ultra-high purity SUS316L. The researches of the two scholars have shown that ultrasonic grinding force is significantly lower than common grinding force. But they failed to analyze the influence of vibration parameters on the grinding force. Wang et al [7] established the model of grinding force of axial ultrasonic vibration assisted grinding ceramic materials based on the relationship of the equations among motion abrasive, the critical load, transverse crack width, longitudinal crack depth of hard and brittle ceramic materials and the rate of material removal. This model was not divided into two parts, cutting deformation force and sliding force, so it failed to definitely indicate the influential degree that cutting deformation force and sliding force affected the grinding force. Zhang [8] conducted a detailed research on the characteristics of one-dimensional ultrasonic vibration assisted grinding and further established a mathematical model of grinding force of the single grain in ultrasonic grinding process. She conducted the study from the cutting deformation force and sliding force and the results manifested that ultrasonic vibrations can reduce grinding force and friction coefficient. This was an experimental research on friction but did not derive a theoretical model of friction. Qin [9] based on the relationship between cutting force of the single grain and maximum contact force, the maximum indentation depth as well as the volume of material removal, derived the calculation expressions of grinding force of constant speed rotary ultrasonic vibration assisted grinding for ductile material, hard and brittle material. But they just discussed the influence of grinding parameters and neglected the effects of vibration parameters on them. Chen et al [10] made a theoretical analysis of grinding force on ultrasonic grinding ceramic from a macro-mechanical perspective and analyzed the reasons for the decrease of grinding force. The study showed that grinding force decreased with the increase of ultrasonic frequency, but it decreased slowly when the frequency exceeded a certain value. But they did not analyze the effect of amplitude on the grinding force.
To sum up, the ultrasonic vibration grinding force on modeling studies has not evolved into a complete system. With so few studies on analysis of ultrasonic grinding force from the point of deformation of cutting force and friction as well as rarely-reported friction modeling study, this work made a systematical research on ultrasonic vibration assisted grinding force and built a mathematical model which can predict the grinding force of the ultrasonic grinding force through theoretical and experimental integrated modeling approach.
2 Theoretical modeling of axial ultrasonic vibration grinding force
Werner [11] and Li and FU [12] established a grinding model as
(1)
where Fn is the normal grinding force; Ft is the tangential grinding force; Fnc and Ftc represent normal chip formation force and the tangential chip formation force produced due to cutting deformation, respectively; Fns and Fts represent the normal sliding force and the tangential sliding force produced due to friction, respectively.
2.1 Cutting deformation force modeling of AUVAG
Kumabe [13] systematically studied vibration cutting techniques and mechanisms, and said that a spatial sine curve appeared in the cutting trajectory of abrasive on the surface of grinding wheel, as shown in Fig. 1.
The single grain trajectory length in grinding zone during common surface grinding process can be written as [14]
Fig. 1 single abrasive grain trajectory of axial ultrasonic vibration assisted grinding
(2)
where Vs is velocity of the grinding wheel; Vw is velocity of the workpiece feed; ap is grinding depth; and ds is grinding wheel diameter.
in the case of same grinding parameters, the single grain trajectory length in single rotation period of common grinding (CG) and ultrasonic grinding can be respectively defined as
(3)
(4)
where T is the single rotation period of ultrasonic vibration, and T=1/f, f is the ultrasonic frequency; Vx is velocity of the single grain in cutting direction; Vy is velocity of the single grain in the radial direction of the grinding wheel; Vz is velocity of the single grain in the axial direction of the grinding wheel; ωs is the rotation angular velocity of wheel; A is ultrasonic amplitude; ω is grinding wheel angular velocity; φ is ultrasonic vibration initial phase, assuming that φ=0.
According to the squeeze rule, it can be computed as
(5)
Since Vs>>Vw, and approximated to laT≈(Vs+Vw)T+ 4A, so the trajectory length of single grain in grinding zone of ultrasonic grinding is
(6)
where n is the number of vibration of single grain in the grinding zone, n=Δt/T=(apds)1/2/(VsT).
The arc length ratio of CG and AUVAG of single grain in the grinding zone can be obtained as
(7)
The cutting force model of the single grain of AUVAG is shown in Fig. 2 and the cutting force of the single grain is
(8)
where fd is the angle between the movement direction of the abrasive and the tangential direction of the grinding wheel, namely fd=arctan(Aωcos(ωt+φ)/Vs); ψ1 and ψ2 are respectively start angle and end angle of contact of the abrasive and workpiece, and during common grinding, ψ1=-π/2 and ψ2=π/2, as to UVAG, ψ1=-π/2+fd and ψ2=π/2+fd; Fp is the unit grinding force; aag is average thickness of undeformed chip of AUVAG.
Fig. 2 cutting force model of single grain in axial ultrasonic vibration assisted grinding:
Since the eccentric deviation can be decomposed into two independent random variables at axial direction and radial direction which both obey normal distribution with zero mean value and equal variance value, it is reasonable to presume that the distribution of the grains on the grinding wheel obey Rayleigh distribution. Then, supposing that each undeformed chip is idealized as a long slab with a uniform thickness h, has an increasing chip thickness from zero to a maximum value hm with a cross section determined by the grain geometry, the undeformed chip thickness obeys a Rayleigh distribution, each grain is regular cone with an internal angle of 2θ, all protruding grains participate in cutting and removing material, no lateral ridge for workpiece material occurs during ultrasonic machining with undeformed workpiece and the grinding wheel. Therefore, the undeformed chip thickness of AUVAG is expressed as [15]
(9)
where E(h) is the expected value of the Rayleigh distribution; h is triangular cross-sectional height of the undeformed chip; C is the number of effective abrasive blades in unit area of grinding wheel.
Equations (7) and (9) are substituted to Eq. (8), and then the grain number through the dynamic grinding area in grinding zone is multiplied, Nd=blpC, b is the grinding wheel width [16], and constant values λ1 and λ2 are respectively added in order to simplify the model; the above analysis is based on the ideal model of cutting deformed force of average thickness of undeformed chip with friction unconsidered. Therefore, the tangential force and normal force reduced by cutting deformed during ultrasonic vibration grinding can be established as
(10)
2.2 Friction force modeling of AUVAG
In the conventional surface grinding process, the grinding force generated by friction is [17]
(11)
where δ is the top area of single grain which participates in cutting; 
is the average contact press between the actual wear surface of abrasive grains and workpiece [18]; μ is the friction coefficient.
The equation of the friction coefficient during AUVAG reads
(12)
where r is the friction coefficient ratio of AUVAG to CG; μp is the friction coefficient of CG, namely μp= αdsVs/(4p0Vw)+ζ; p0 is a constant determined by experiment; α and ζ are coefficients determined by physical and mechanical properties of contact interface, respectively [19].
Therefore, the friction model of AUVAG is established after adding constant yields λ3 and λ4 in the model by substituting Eq. (12) into Eq. (11), and taking into consideration the friction coefficient affected by the workpiece material, grit size, grinding parameters and vibration parameters, etc.,
(13)
2.3 Grinding force modeling of AUVAG
By all accounts, since Vs>>Vw, 
and r are regarded as constant. And it is generally agreed that Fp is in association with workpiece material and model of grinding wheel, θ, b, δ and C with the model of grinding wheel, α with workpiece material. Then, associating Eq. (10) with Eq. (13), the grinding force of AUVAG can be further derived as
(14)
where K1=π2Fpbcosθ/16, K2=rαCδb, K3=4rCζδp0b, K4= πFpbsinθcosfd/4, K5=4Cδp0b, λ0=λ1+λ3, 
3 Experimental modeling of AUVAG force
3.1 Experiment principle and equipment
The experiment of dry infeed-grinding of 45# steel is conducted on M7130 surface grinding machine with horizontal grinding wheel spindle and reciprocating table. Al2O3 grinding wheel is dressed by a single point diamond pen. Grinding force is measured by SDC-C4F universal strain-gauge three-component-dynamometer. Experiment parameters: wheel diameter ds=350 mm, wheel velocity Vs=26 m/s, workpiece velocity Vw=160, 240, 320 mm/s, grinding depth ap=5, 10, 15, 20 μm, ultrasonic generator’s frequency f=20 kHz, and transducer’s amplitude A=6, 9, 12 μm, the sampling frequency of dynamometer is 60 Hz. Comparative test method is adopted in order to study the grinding force of AUVAG well: when the ultrasonic generator is open, ultrasonic grinding works; and when off, conventional grinding works. Operating principle of axial ultrasonic vibration assisted grinding can be seen in Fig. 3.
3.2 Modeling and prediction of regression equation of ultrasonic vibration grinding force
Based on the MATLAB software, the regression equation (Eq. (15)) is derived after regression analysis of substituting the experimental data in Table 1 into Eq. (14). Effectiveness analysis of the equation is shown in Table 2 which proves distinctness of the regression equation. The residential plot which says experimental value and computed value of grinding force is shown in Fig. 4. the ledgements mean the residual confidence intervals, the filled dots represent the actual residual area. If the confidence intervals pass through the origin, it is indicated that the regression line of grinding force fits better, or the point is outlier. The sequence number of regression analysis data is regarded as horizontal ordinate, and the residual is vertical ordinate, while line segments are 95% of residual confidence intervals between the experimental value and the computed value, and the filled dots are residual means. The closer the filled dots come near zero, the better the regression line of grinding force fits the experimental values. According to the residual plot, in addition to the fact that the 14th dot of normal grinding force and the 12th dot of tangential grinding force do not pass the origin, the other dots all pass the origin, which demonstrates that the regression line of grinding force better fits with the experimental values. Thus, it further indicates that the force model of AUVAG built by comprehensive modeling is feasible.
(15)
Fig. 3 Operating principle of axial ultrasonic vibration assisted grinding
Table 1 Regression analysis data of ultrasonic grinding force
Table 2 Effectiveness analysis results of regression equation
Based on the mathematical model of ultrasonic grinding force, the surface chart of relationship among the speed of the workpiece, the grinding force and grinding depth is shown in Fig. 5 under the circumstances that the wheel speed is 26 m/s, the amplitude is 12 μm; while the surface chart of relationship among vibration frequency, amplitude and grinding force is shown in Fig. 6 under the circumstances that the wheel speed is 26 mm/s and grinding depth is 10 μm. Comparative analysis showed that when the amplitude is certain, workpiece velocity and grinding depth are two significant factors affecting tangential and normal grinding forces, and both tend to increase with the increase of tangential and normal grinding forces; when the grinding depth is certain, vibration frequency and amplitude are two factors affecting tangential and normal grinding forces, and both decrease with the increase of tangential and normal grinding forces because the larger the amplitude is, the more obvious the separation of abrasive grains and the workpiece would be, the shorter the time of the unit grinding force would act on the workpiece, the lower the average cutting force would be. Therefore, during the process of AUVAG, the grinding force is mainly affected by action of wheel velocity, workpiece velocity, grinding depth, amplitude, frequency and other factors. And grinding parameters and ultrasonic vibration parameters should be reasonably matched in order to achieve desirable grinding forces.
Fig. 4 Residual plot of modeling of ultrasonic grinding force:
Fig. 5 surface chart of relationship among workpiece, velocity, grinding depth and grinding force:
To further verify accuracy of the regression equation, Eq. (15) is used to predict the value of the grinding force in the case of other parameters. The experiment and prediction results are listed in Table 3 and the error graph is shown in Fig. 7. According to Table 3, the maximum relative error between normal grinding force of experimental values and predictive values is 11.47% and its average relative error is 5.41%; while the maximum relative error of the tangential grinding force is 10.14% and its average relative error is 4.29%. As shown in fig. 7, the relative error falls within a reasonable limit. Therefore, the model is proved to be accurate and reliable.
Fig. 6 surface chart of relationship among frequency, amplitude and grinding force:
Table 3 Predicted results of regression analysis
Fig. 7 Forecast errors chart of regression equation of ultrasonic grinding force:
4 Conclusions
1) a regression analysis on the theoretical model is made by using the experimental data and the regression coefficients of the theoretical model of ultrasonic vibration grinding force are computed. the significant test is conducted and the mathematical model of grinding force is established. the model is employed to make prediction of the grinding force which shows that the grinding force decreases when the frequency and amplitude increase and the grinding force increases with the increase of grinding depth and workpiece velocity. The maximum relative error between predicted values and experimental values of normal grinding force is 11.47% and its average relative error is 5.41%. The maximum relative error of the tangential grinding force is 10.14% and its average relative error is 4.29%. What’s more, the relative error figure shows that predictive values agreed well with experimental values. Thus, it proved reliability of the model.
2) An integrated modeling approach was adopted to construct surface chart of relationship among frequency, amplitude and grinding force when the wheel velocity is 26 m/s and amplitude is 12 μm. And the surface chart of relationship among frequency, amplitude and grinding force is also obtained when the wheel velocity is 26 mm/s and grinding depth is 10 μm. The surface chart showed that the grinding force was greatly affected by workpiece velocity, grinding depth, frequency, amplitude and other factors. Hence, consistency of the law with the theoretical analysis in the research indicates accuracy of the model.
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(Edited by DENG Lü-xiang)
Foundation item: Project(51275530) supported by the National Natural Science Foundation of China
Received date: 2014-12-20; Accepted date: 2015-11-06
Corresponding author: He Yu-hui, PhD, associate professor; Tel: +86-18975837931; E-mail: csuhyh@163.com
