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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 927876, 28 pages
http://dx.doi.org/10.1155/2011/927876
Research Article

Exact Solution for the Time-Dependent Temperature Field in Dry Grinding: Application to Segmental Wheels

1Cátedra Energesis de Tecnología Interdisciplinar, Universidad Católica de Valencia, 46002 Valencia, Spain
2Departamento de Matemáticas, Universidad de Pinar del Río, 20200 Pinar del Río, Cuba
3Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain

Received 21 February 2011; Accepted 1 April 2011

Academic Editor: Ezzat G. Bakhoum

Copyright © 2011 J. L. González-Santander et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a closed analytical solution for the time evolution of the temperature field in dry grinding for any time-dependent friction profile between the grinding wheel and the workpiece. We base our solution in the framework of the Samara-Valencia model Skuratov et al., 2007, solving the integral equation posed for the case of dry grinding. We apply our solution to segmental wheels that produce an intermittent friction over the workpiece surface. For the same grinding parameters, we plot the temperature fields of up- and downgrinding, showing that they are quite different from each other.

1. Introduction

A major technological challenge in the grinding of metallic plates [15] is how to avoid thermal damage to the workpiece. The grinding process transforms large amounts of mechanical energy into heat, which primarily affects the contact area between the workpiece and the wheel. It is therefore of great industrial importance to determine the temperature distribution within the workpiece, and its maximum, in order to avoid thermal damage.

Despite the fact that there have been studies of the temperature field solving the heat equation numerically [6, 7], an analytical approach is of great interest [8] for two reasons. Firstly, explicit expressions for the dependence of the temperature field with respect to the grinding parameters can be obtained. Secondly, the rapid presentation of results allows the industry to monitor the grinding process on line.

This paper is organized as follows. Section 2 presents the Samara-Valencia model [9]. This model is used in Section 3 to derive a closed analytical solution for the evolution of the temperature field in dry grinding, for any time-dependent friction profile between the grinding wheel and the workpiece. Section 4 applies the result obtained in the previous section to intermittent grinding, for both up- and downgrindings. Section 5 analyzes some important variables in continuous grinding, such as the location of the maximum temperature and the relaxation time, which can be applied to intermittent grinding. We compare also the stationary regime of continuous grinding with the quasistationary regime of intermittent grinding. In Section 6, we present some numerical results, comparing continuous and intermittent grinding. Our conclusions are summarized in Section 7.

2. Samara-Valencia Model

The Samara-Valencia model setup is depicted in Figure 1. The workpiece moves at a constant speed 𝑣𝑑 and is assumed to be infinite along 𝑂𝑥 and 𝑂𝑧, and semiinfinte along 𝑂𝑦. The plane 𝑦=0 is the surface being ground. The contact area between the wheel and the workpiece is an infinitely long strip of width 𝛿 located parallel to the 𝑂𝑧 axis and on the plane 𝑦=0. Both the wheel and the workpiece are assumed to be rigid. Although the equations below allow for the case of wet grinding, we will consider in this paper the case of dry grinding. The Samara-Valencia model [9] solves the convection heat equation𝜕𝑡𝑇𝜕(𝑡,𝑥,𝑦)=𝑘𝑥𝑥𝑇(𝑡,𝑥,𝑦)+𝜕𝑦𝑦𝑇(𝑡,𝑥,𝑦)𝑣𝑑𝜕𝑥𝑇(𝑡,𝑥,𝑦),(2.1) subject to the initial condition,𝑇(0,𝑥,𝑦)=0,(2.2) and the boundary condition,𝑘0𝜕𝑦𝑇(𝑡,𝑥,0)=𝑏(𝑡,𝑥)𝑇(𝑡,𝑥,0)+𝑑(𝑡,𝑥),(2.3) where <𝑥<and 𝑡,𝑦0. The first term of (2.3) models the application of coolant over the workpiece surface considering 𝑏(𝑡,𝑥) as the heat transfer coefficient. The second term, 𝑑(𝑡,𝑥), represents the heat flux entering into the workpiece. This heat flux is generated on the surface by friction between the wheel and the workpiece. The solution of the Samara-Valencia model (2.1)–(2.3) may be presented as the sum of two terms,𝑇(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑇(1)(𝑡,𝑥,𝑦),(2.4) where𝑇(0)1(𝑡,𝑥,𝑦)=4𝜋𝑘0𝑡0𝑑𝑠𝑠exp𝑦24𝑘𝑠𝑑𝑥𝑑𝑡𝑠,𝑥𝑥exp𝑥𝑣𝑑𝑠2,𝑇4𝑘𝑠(2.5)(1)(1𝑡,𝑥,𝑦)=4𝜋𝑡0𝑑𝑠𝑠exp𝑦24𝑘𝑠𝑑𝑥𝑦𝑏2𝑘𝑠𝑡𝑠,𝑥𝑘0×𝑇𝑡𝑠,𝑥𝑥,0exp𝑥𝑣𝑑𝑠2.4𝑘𝑠(2.6) Notice that 𝑇(0) contains the friction function 𝑑(𝑡,𝑥), and 𝑇(1) contains the temperature field on the surface and the heat transfer coefficient 𝑏(𝑡,𝑥).

927876.fig.001
Figure 1: Bidimensional model for flat grinding.

3. 𝑇(0) Theorem for Dry Grinding

3.1. Dry Grinding

When no coolant is applied to the workpiece, we can consider the workpiece to be isolated from the environment. According to Newton's cooling law, this means that there is no heat flux from the workpiece to the environment, thus the heat transfer coefficient is zero,𝑏(𝑡,𝑥)=0.(3.1) In this case of dry grinding, the expression for 𝑇(1) given in (2.6) becomes𝑇(1)𝑦(𝑡,𝑥,𝑦)=8𝜋𝑘𝑡0exp𝑦24𝑘𝑠𝑑𝑠𝑠2×𝑑𝑥𝑇𝑡𝑠,𝑥𝑥,0exp𝑥𝑣𝑑𝑠2.4𝑘𝑠(3.2) In order to tackle the integral equation given in (3.2), let us define the following integral operators𝑇(1)(𝑡,𝑥,𝑦)=𝑦[],𝑇(𝑡,𝑥,0)(3.3) where𝑦[]𝑦𝑇(𝑡,𝑥,0)=8𝜋𝑘𝑡0𝑑𝑠𝑠2exp𝑦24𝑘𝑠𝑠[],𝑇(𝑡,𝑥,0)(3.4)𝑠[𝑇](𝑡,𝑥,0)=𝑑𝑥𝑇𝑡𝑠,𝑥𝑥,0exp𝑥𝑣𝑑𝑠2.4𝑘𝑠(3.5) Therefore, taking into account (3.3), we may rewrite (2.4) as𝑇(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦[].𝑇(𝑡,𝑥,0)(3.6)

3.2. The 𝑠 Operator

Let us calculate the 𝑠 operator over the frictional term 𝑇(0) of the temperature field. According to (2.5), 𝑇(0) may be expressed as𝑇(0)𝑡𝑠,𝑥=,014𝜋𝑘00𝑡𝑠𝑑𝜎𝜎𝑑𝜉𝑑(𝑡𝑠𝜎,𝜉)exp𝜉𝑥𝑣𝑑𝜎2.4𝑘𝜎(3.7) Therefore, substituting (3.7) in (3.5), and reordering the integrals by Fubini’s theorem, we obtain𝑠𝑇(0)(=𝑡,𝑥,0)14𝜋𝑘0𝑑𝜉0𝑡𝑠𝑑𝜎𝜎×𝑑(𝑡𝑠𝜎,𝜉)𝑑𝑥𝑥exp𝑥𝑣𝑑𝑠24𝑘𝑠𝜉𝑥𝑣𝑑𝜎2.4𝑘𝜎(3.8) Expanding the exponent of the integrand given in (3.8), we arrive at𝑠𝑇(0)=(𝑡,𝑥,0)14𝜋𝑘0exp𝑥+𝑣𝑑𝑠2×4𝑘𝑠𝑣𝑑𝜉exp𝑑𝜉×2𝑘0𝑡𝑠𝑑𝜎𝜎𝜉𝑑(𝑡𝑠𝜎,𝜉)exp2𝑣4𝑘𝜎2𝑑𝜎×4𝑘𝑑𝑥𝑥exp24𝑘𝑠+𝜎+𝑥𝑠𝜎2𝑘𝑥𝜎+𝜉𝑠.𝑠𝜎(3.9) The last integral given in (3.9) can be calculated [10, Equation 3.323.2], so that,𝑠𝑇(0)=(𝑡,𝑥,0)𝑘𝑠2𝜋𝑘0exp𝑥+𝑣𝑑𝑠2×4𝑘𝑠𝑣𝑑𝜉exp𝑑𝜉×2𝑘0𝑡𝑠𝑑𝜎𝑑(𝑡𝑠𝜎,𝜉)𝜎𝑠+𝜎exp(𝑥𝜎+𝜉𝑠)2𝜉4𝑘𝑠𝜎(𝑠+𝜎)2𝑣4𝑘𝜎2𝑑𝜎.4𝑘(3.10) Once again, expanding the exponent of the last integrand given in (3.10) and simplifying, we arrive at𝑠𝑇(0)=(𝑡,𝑥,0)𝑘𝑠2𝜋𝑘0exp𝑥𝑣𝑑2𝑘𝑣𝑑𝜉exp𝑑𝜉×2𝑘0𝑡𝑠𝑑𝜎𝑑(𝑡𝑠𝜎,𝜉)𝜎𝑠+𝜎exp(𝑥𝜉)2𝑣4𝑘(𝑠+𝜎)2𝑑(.4𝑘𝜎+𝑠)(3.11) Let us define𝐼𝜎=0𝑡𝑠𝑑𝜎𝑑(𝑡𝑠𝜎,𝜉)𝜎𝑠+𝜎exp(𝑥𝜉)2𝑣4𝑘(𝑠+𝜎)2𝑑.4𝑘(𝜎+𝑠)(3.12) We can calculate (3.12) performing the substitution, 𝜇=𝜎+𝑠, and introducing the Heaviside function 𝐻(𝑥), so that,𝐼𝜎=𝑡0𝑑𝜇𝑑(𝑡𝜇,𝜉)𝜇𝑠𝜇exp(𝑥𝜉)2𝑣4𝑘𝜇2𝑑𝜇4𝑘𝐻(𝜇𝑠).(3.13) Substituting (3.13) in (3.11) and simplifying, we get𝑠𝑇(0)=(𝑡,𝑥,0)𝑘𝑠2𝜋𝑘0𝑑𝜉𝑡0𝑑𝜇𝑑(𝑡𝜇,𝜉)𝜇𝑠𝜇exp𝜉𝑥𝑣𝑑𝜇24𝑘𝜇𝐻(𝜇𝑠).(3.14)

3.3. The 𝑦 Operator

Substituting the expression obtained in (3.14) into (3.4) and reordering the integrals, we have𝑦𝑇(0)=(𝑡,𝑥,0)𝑦16𝜋3/2𝑘0𝑘×𝑑𝜉𝑡0𝑑𝜇𝑑(𝑡𝜇,𝜉)𝜇exp𝜉𝑥𝑣𝑑𝜇2×4𝑘𝜇𝑡0𝑑𝑠𝐻(𝜇𝑠)𝑠3/2𝜇𝑠exp𝑦2.4𝑘𝑠(3.15) Let us define𝐼𝑠=𝑡0𝑑𝑠𝐻(𝜇𝑠)𝑠3/2𝜇𝑠exp𝑦2.4𝑘𝑠(3.16) Since 𝜇[0,𝑡], the integral given in (3.16) can be expressed in the following way:𝐼𝑠=𝜇0𝑑𝑠𝑠3/2𝜇𝑠exp𝑦2.4𝑘𝑠(3.17) In order to calculate (3.17), we can perform the following substitutions: 𝑠=𝜇/𝑡, 𝑤=𝑡1 and 𝑟=𝑦𝑤/2𝑘𝜇, leading to𝐼𝑠=2𝜋𝑘𝑦𝜇𝑦exp24𝑘𝜇.(3.18) Therefore, substituting (3.18) in (3.15) and changing the integration order, we arrive at 𝑦𝑇(0)=(𝑡,𝑥,0)18𝜋𝑘0𝑡0𝑑𝜇𝜇𝑦exp24𝑘𝜇𝑑𝜉𝑑(𝑡𝜇,𝜉)exp𝑥𝜉𝑣𝑑𝜇2.4𝑘𝜇(3.19) Remembering the expression for 𝑇(0) given in (2.5), we conclude𝑦𝑇(0)=1(𝑡,𝑥,0)2𝑇(0)(𝑡,𝑥,𝑦).(3.20)

3.4. Resolution by Successive Approximations

According to (3.2), in order to evaluate 𝑇(1), we have to know the temperature field on the surface, 𝑇(𝑡,𝑥,0). At zeroth order approximation, 𝑇0, we can consider that the temperature field will be given by the term involving friction only, that is, 𝑇(0) according to (2.5). So that,𝑇0(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦).(3.21) In order to get the first-order approximation 𝑇1, we can substitute the zeroth order (3.21) in (3.3), 𝑇1(1)(𝑡,𝑥,𝑦)=𝑦𝑇0(𝑡,𝑥,0)=𝑦𝑇(0).(𝑡,𝑥,0)(3.22) Thus, the temperature field at first order is𝑇1(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑇1(1)(𝑡,𝑥,𝑦),(3.23) or according to (3.22),𝑇1(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦𝑇0.(𝑡,𝑥,0)(3.24) In general, the 𝑛th (𝑛=0,1,2,) approximation is𝑇𝑛(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦𝑇𝑛1,(𝑡,𝑥,0)(3.25) where the initial value is given by (3.21). Applying now (3.20) to (3.22), we can rewrite the first-order approximation as𝑇13(𝑡,𝑥,𝑦)=2𝑇(0)(𝑡,𝑥,𝑦).(3.26) In order to evaluate the second order, we can substitute (3.26) in the recurrence equation (3.25) for 𝑛=2. Taking into account that the integral operator 𝑦 is linear, we obtain𝑇2(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦𝑇1(𝑡,𝑥,0)=𝑇(0)3(𝑡,𝑥,𝑦)+2𝑦𝑇(0)=7(𝑡,𝑥,0)4𝑇(0)(𝑡,𝑥,𝑦),(3.27) where we have applied (3.20) once again. Repeating the same steps, we get at third order𝑇3(𝑡,𝑥,𝑦)=158𝑇(0)(𝑡,𝑥,𝑦).(3.28) Looking at the coefficients appearing in the first orders, (3.26), (3.27), and (3.28), we may establish the following conjecture for the 𝑛th order:𝑇𝑛2(𝑡,𝑥,𝑦)=𝑛+112𝑛𝑇(0)(𝑡,𝑥,𝑦),(3.29) that can be proved by induction, 𝑇𝑛+1(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦𝑇𝑛(𝑡,𝑥,0)=𝑇(0)2(𝑡,𝑥,𝑦)+𝑛+112𝑛𝑦𝑇(0)=2(𝑡,𝑥,0)𝑛+212𝑛+1𝑇(0)(𝑡,𝑥,𝑦).(3.30) The temperature field will be the infinite order approximation, thus taking the limit of (3.29), results in𝑇(𝑡,𝑥,𝑦)=lim𝑛𝑇𝑛(𝑡,𝑥,𝑦)=2𝑇(0)(𝑡,𝑥,𝑦).(3.31) Applying (3.20), we may check that (3.31) is a solution of the integral equation given in (3.6), 𝑇(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦[]𝑇(𝑡,𝑥,0)=𝑇(0)(𝑡,𝑥,𝑦)+2𝑦𝑇(0)(𝑡,𝑥,0)=2𝑇(0)(𝑡,𝑥,𝑦).(3.32) Taking into account (2.5), we conclude that the time evolution of the temperature field may be expressed as1𝑇(𝑡,𝑥,𝑦)=2𝜋𝑘0𝑡0𝑑𝑠𝑠exp𝑦24𝑘𝑠𝑑𝑥𝑑𝑡𝑠,𝑥𝑥exp𝑥𝑣𝑑𝑠2.4𝑘𝑠(3.33)

3.5. Uniqueness of the Solution
3.5.1. Bound Limit for 0

To prove the uniqueness of the solution of the integral equation (3.6), let us calculate first the value of the 𝑦 operator over a constant. According to (3.5), we have𝑠[1]=𝑥exp𝑥𝑣𝑑𝑠24𝑘𝑠𝑑𝑥.(3.34) Performing the substitution: 𝑢=(𝑥𝑥𝑣𝑑𝑠)/2𝑘𝑠, (3.33) results in𝑠[1]=2𝑘𝑠𝑒𝑢2𝑑𝑢=2𝜋𝑘𝑠.(3.35) Applying (3.35) to (3.4), we have 𝑦[1]=𝑦4𝜋𝑘𝑡0𝑑𝑠𝑠3/2exp𝑦2.4𝑘𝑠(3.36) Performing the substitution: 𝑢=𝑦/2𝑘𝑠, we have𝑦[1]=1𝜋𝑦/2𝑘𝑡𝑒𝑢21𝑑𝑢=2𝑦erfc2𝑘𝑡.(3.37) Therefore,0[1]=12.(3.38)

Let us consider now a function 𝑓(𝑡,𝑥,𝑦) whose maximum value taking 𝑦=0 is 𝑓max, that is,𝑓(𝑡,𝑥,0)𝑓max.(3.39) Applying 0 to (3.39) and taking into account that 0 is a linear operator,0[𝑓](𝑡,𝑥,0)0𝑓max=𝑓max0[1].(3.40) Thus, according to (3.38),0[]𝑓𝑓(𝑡,𝑥,0)max2.(3.41) Note that, if we apply 0 to (3.39) and we take into account (3.37), we have 0(2)[]𝑓(𝑡,𝑥,0)0𝑓max2=𝑓max22.(3.42) So, in general, for all 𝑛,0(𝑛)[]𝑓𝑓(𝑡,𝑥,0)max2𝑛.(3.43)

3.5.2. Resolution of the Uniqueness

If 𝑇𝐴(𝑡,𝑥,𝑦) and 𝑇𝐵(𝑡,𝑥,𝑦) are solutions of (3.6), we have𝑇𝐴(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦𝑇𝐴,𝑇(𝑡,𝑥,0)(3.44)𝐵(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦𝑇𝐵.(𝑡,𝑥,0)(3.45) Subtracting (3.45) from (3.44) and taking into account that 𝑦 is a linear operator,𝑇𝐴(𝑡,𝑥,𝑦)𝑇𝐵(𝑡,𝑥,𝑦)=𝑦𝑇𝐴(𝑡,𝑥,0)𝑇𝐵.(𝑡,𝑥,0)(3.46) Taking 𝑦=0 in (3.46),𝑇𝐴(𝑡,𝑥,0)𝑇𝐵(𝑡,𝑥,0)=0𝑇𝐴(𝑡,𝑥,0)𝑇𝐵(𝑡,𝑥,0).(3.47) Recursive substitution of (3.47) yields𝑇𝐴(𝑡,𝑥,0)𝑇𝐵(𝑡,𝑥,0)=0(𝑛)𝑇𝐴(𝑡,𝑥,0)𝑇𝐵(𝑡,𝑥,0).(3.48) If we take in (3.43) as a function 𝑓, 𝑓1(𝑡,𝑥,0)=𝑇𝐴(𝑡,𝑥,0)𝑇𝐵(𝑡,𝑥,0),(3.49) according to (3.48), we have that, for all 𝑛,𝑇𝐴(𝑡,𝑥,0)𝑇𝐵𝑓(𝑡,𝑥,0)1,max2𝑛,(3.50) where 𝑓1,max is the maximum value of 𝑓1(𝑡,𝑥,0). Taking the limit in (3.50), 𝑇𝐴(𝑡,𝑥,0)𝑇𝐵(𝑡,𝑥,0)lim𝑛𝑓1,max2𝑛=0,(3.51) so that,𝑇𝐴(𝑡,𝑥,0)𝑇𝐵(𝑡,𝑥,0).(3.52) Note that in (3.48) we can exchange labels 𝐴 and 𝐵, 𝑇𝐵(𝑡,𝑥,0)𝑇𝐴(𝑡,𝑥,0)=0(𝑛)𝑇𝐵(𝑡,𝑥,0)𝑇𝐴.(𝑡,𝑥,0)(3.53) Thus, taking now the function 𝑓2(𝑡,𝑥,0)=𝑇𝐵(𝑡,𝑥,0)𝑇𝐴(𝑡,𝑥,0),(3.54) we obtain that 𝑇𝐵(𝑡,𝑥,0)𝑇𝐴(𝑡,𝑥,0)lim𝑛𝑓2,max2𝑛=0,(3.55) that is,𝑇𝐵(𝑡,𝑥,0)𝑇𝐴(𝑡,𝑥,0).(3.56) From (3.52) and (3.56), we conclude that both solutions on the surface are equal,𝑇𝐴(𝑡,𝑥,0)=𝑇𝐵(𝑡,𝑥,0).(3.57) Applying 𝑦 to (3.57),𝑦𝑇𝐴(𝑡,𝑥,0)=𝑦𝑇𝐵,(𝑡,𝑥,0)(3.58) and substituting (3.58) in (3.44), we have that𝑇𝐴(𝑡,𝑥,𝑦)=𝑇(0)(𝑡,𝑥,𝑦)+𝑦𝑇𝐵.(𝑡,𝑥,0)(3.59) Comparing (3.45) with (3.59), we finally obtain 𝑇𝐴(𝑡,𝑥,𝑦)=𝑇𝐵(𝑡,𝑥,𝑦).(3.60) Therefore, the solution given in (3.33) is the only solution of (3.6).

4. Intermittent Grinding

Equation (3.31) is a generalization of the result presented in [11] since now the transient regime is considered and any type of time-dependent friction profile is allowed. In the next section, we will apply (3.31) to calculate the time-dependent temperature field produced by an intermittent grinding of a segmental wheel (Figure 2).

927876.fig.002
Figure 2: Profile of a toothed wheel.
4.1. Intermittence Function

Let us model the friction due to a toothed wheel, which can contact the workpiece within 𝑥[0,𝛿]. Therefore, we will call this zone, contact zone. Figures 3 and 4 show the friction zone highlighted in red within the limits 𝑎 and 𝑏 for two different times 𝑡1 and 𝑡2>𝑡1. The wheel has a spatial period 𝜒=𝜒0+𝜒1, where 𝜒0 is the distance between teeth and 𝜒1 is the tooth width. The wheel teeth move at a speed 𝑣𝑚=𝜔𝑅, where 𝜔 is the angular velocity and 𝑅 is the wheel radius. When more than two teeth touch simultaneously the contact zone [0,𝛿], the friction zone is split as Figure 5 shows. For a given instant 𝑡, the incoming heat flux 𝑑(𝑡,𝑥) enters the workpiece through the friction zone: 𝑥(𝑎𝑗,𝑏𝑗), 𝑗=0,,𝑛+1, where 𝑗 indicates a wheel tooth. Notice that there can be up to 𝑛+2 teeth within the contact zone, where𝛿𝑛=𝜒.(4.1) Note, also, that the friction limits are time dependent: 𝑎𝑗=𝑎𝑗(𝑡) and 𝑏𝑗=𝑏𝑗(𝑡). If the incoming heat flux 𝑞 is constant for every point where friction occurs, we may write the friction function as𝑑(𝑡,𝑥)=𝑞𝑛+1𝑗=0𝐻𝑥𝑎𝑗𝐻𝑏(𝑡)𝑗,(𝑡)𝑥(4.2) where 𝐻(𝑥) is the Heaviside function. In order to know the friction limits of the wheel teeth (𝑗=0,,𝑛+1) which enters into the contact zone, that is, 𝑎𝑗 and 𝑏𝑗, let us define the spatial period,𝜒=(𝑛+2)𝜒.(4.3) According to Figure 5, the 𝑔𝑗(𝑡) points, 𝑗=0,,𝑛+1, are initially over the period 𝜒,𝑔𝑗(𝑡)=(1)𝜙𝑣𝑚𝑡+𝜒𝑗𝜒,(4.4) where we have defined a boolean variable 𝜙, in order to define the rotation of the wheel: 𝜙=1, downgrinding; 𝜙=0, upgrinding, as Figure 6 shows. If we want a periodic repetition of the friction limits over the period 𝜒, we may define the function𝑓𝑗(𝑡)=𝑔𝑗𝑔(𝑡)𝑗(𝑡)𝜒𝜒.(4.5) We want as well that 𝑏𝑗[0,𝛿], thus,𝑏𝑗𝑓(𝑡)=minmax𝑗=𝑏(𝑡),0,𝛿𝑗(𝑡)0<𝑏𝑗(𝑡)<𝛿,0𝑏𝑗(𝑡)0,𝛿𝑏𝑗(𝑡)𝛿.(4.6) Similarly, since the tooth width is 𝜒1,𝑎𝑗𝑓(𝑡)=minmax𝑗(𝑡)𝜒1.,0,𝛿(4.7) The min and max functions are given by ||||min(𝑎,𝑏)=𝑎+𝑏𝑎𝑏2,||||max(𝑎,𝑏)=𝑎+𝑏+𝑎𝑏2.(4.8)

927876.fig.003
Figure 3: Friction function 𝑑(𝑡,𝑥) for time 𝑡1 highlighted in red.
927876.fig.004
Figure 4: Friction function 𝑑(𝑡,𝑥) for time 𝑡2>𝑡1 highlighted in red.
927876.fig.005
Figure 5: Initial location of the 𝑔𝑗(𝑡) points, for 𝑛=2.
927876.fig.006
Figure 6: Upgrinding and downgrinding according to the wheel rotation.
4.2. Temperature Field

Substituting (4.2) in (3.33), we obtain𝑞𝑇(𝑡,𝑥,𝑦)=2𝜋𝑘0𝑡0𝑑𝑠𝑠exp𝑦2×4𝑘𝑠𝑛+1𝑗=0𝑏𝑗𝑎(𝑡𝑠)𝑗(𝑡𝑠)𝑑𝑥𝑥exp𝑥𝑣𝑑𝑠2.4𝑘𝑠(4.9) Let us evaluate the integral over the 𝑥 variable in (4.9),𝐼𝑥=𝑛+1𝑗=0𝑏𝑗𝑎(𝑡𝑠)𝑗(𝑡𝑠)𝑥exp𝑥𝑣𝑑𝑠24𝑘𝑠𝑑𝑥.(4.10) Performing the substitution,𝑥𝑢=𝑥𝑣𝑑𝑠2,𝑘𝑠(4.11) and taking into account the properties of the error function, we get𝐼𝑥=𝜋𝑘𝑠ERF(𝑡,𝑥,𝑠),(4.12) where we have defined the functionERF(𝑡,𝑥,𝑠)=𝑛+1𝑗=0erf𝑥+𝑣𝑑𝑠𝑎𝑗(𝑡𝑠)2𝑣𝑘𝑠erf𝑑𝑠+𝑥𝑏𝑗(𝑡𝑠)2.𝑘𝑠(4.13) Substituting (4.12) in (4.9), we obtain the following expression for the temperature field: 𝑞𝑇(𝑡,𝑥,𝑦)=𝑘2𝜋𝑘0𝑡0𝑑𝑠𝑠exp𝑦24𝑘𝑠ERF(𝑡,𝑥,𝑠).(4.14)

5. Continuous Grinding

5.1. Stationary Regime

In order to calculate the temperature field for the case of continuous friction, we can take in (4.7)-(4.6) the constant values of the contact zone, 𝑎0𝑏(𝑡)=0,0(𝑡)=𝛿.(5.1) Therefore, we can redefine (4.12) as ERFcont(𝑥,𝑠)=erf𝑥+𝑣𝑑𝑠2𝑣𝑘𝑠erf𝑑𝑠+𝑥𝛿2𝑘𝑠,(5.2) obtaining, according to (4.13), the following temperature field:𝑇cont𝑞(𝑡,𝑥,𝑦)=𝑘2𝜋𝑘0𝑡0𝑑𝑠𝑠exp𝑦24𝑘𝑠ERFcont(𝑥,𝑠).(5.3) The stationary regime is reached when the temperature field does not vary in time,𝜕𝑇𝜕𝑡=0.(5.4) In the case of continuous grinding, the time derivative is𝜕𝑇cont(𝑡,𝑥,𝑦)=𝑞𝜕𝑡𝑘2𝑘0ERFcont(𝑥,𝑡)𝜋𝑡exp𝑦24𝑘𝑡.(5.5) Taking the limit of (5.4), knowing that erf(±)=±1, we can check that the stationary regime is reached when 𝑡:lim𝑡ERFcont(𝑥,𝑡)=lim𝑡erf𝑥+𝑣𝑑𝑡2𝑣𝑘𝑡erf𝑑𝑡+𝑥𝛿2𝑘𝑡=0,(5.6) so that,lim𝑡𝜕𝑇cont(𝑡,𝑥,𝑦)𝜕𝑡=0.(5.7)

5.2. Quasistationary Regime

Notice that intermittent grinding never reaches a stationary regime, since the heat source produced by friction is pulsed. This is not the case of continuous grinding, where the stationary regime is reached asymptotically for 𝑡. Therefore, for continuous grinding, we may define a relaxation time 𝑡 that provides us an idea of how rapid the stationary regime is reached in practice. It turns out that this relaxation time, defined for the continuous case, is a good temporal reference in order to plot the temperature field in the case of intermittent grinding. Even though intermittent grinding never reaches a stationary regime, we may define a quasistationary regime in which the temperature field is periodically stable. Since 𝑎𝑗(𝑡) and 𝑏𝑗(𝑡) are periodic functions (4.6)-(4.7), according to (4.14), we may define the quasistationary regime as𝑇𝑞(𝑡,𝑥,𝑦)=𝑘2𝜋𝑘00𝑑𝑠𝑠exp𝑦24𝑘𝑠ERF(𝑡,𝑥,𝑠).(5.8) According to Figure 5, the temporal period of the friction function 𝑑(𝑡,𝑥0) in a fixed point 𝑥=𝑥0 is𝜒𝜏=𝑣𝑚.(5.9) However, the global consideration of the 𝑑(𝑡,𝑥) plot indicates the following temporal period:𝜏=𝜒𝑣𝑚.(5.10) In view of (3.31), we may conclude that 𝑇(𝑡,𝑥,𝑦) possesses the same global and point periods 𝜏 and 𝜏 as the friction function 𝑑(𝑡,𝑥).

5.3. Maximum Temperature

Since the error function erf(𝑧) is an increasing function for all 𝑧, we haveERFcont(𝑥,𝑡)>0𝑡>0,𝑥.(5.11) Therefore, the temperature on a given point (𝑥,𝑦) of the workpiece is a monotonically increasing function,𝜕𝑇cont(𝑡,𝑥,𝑦)𝜕𝑡>0𝑦,𝑡>0,𝑥.(5.12) Equation (5.12) means that the maximum temperature must be reached in the stationary state, 𝑡. Moreover, as in (5.11), we haveERFcont(𝑥,𝑠)>0𝑠(0,𝑡),𝑥,(5.13) so that, for 𝑦>0,𝜕𝑇cont(𝑡,𝑥,𝑦)𝜕𝑦=𝑞𝑦4𝜋𝑘𝑘0𝑡0𝑑𝑠𝑠3/2exp𝑦24𝑘𝑠ERFcont(𝑥,𝑠)<0.(5.14) Equation (5.14) indicates that maximum temperature must be localized on the surface, 𝑦=0. From (5.12) and (5.14), we conclude that the maximum temperature must be reached on the surface in the stationary regime,𝑇max=lim𝑡𝑇cont𝑡,𝑥max.,0(5.15) This result agrees with [11].

5.3.1. Location of the Maximum Temperature

Denoting the stationary regime in the case of continuous friction as 𝑇cont(𝑥,𝑦)=lim𝑡𝑇cont(𝑡,𝑥,𝑦),(5.16) according to [12], we have,𝑇cont(𝑋,𝑌)=𝒯Δ𝑋𝑋𝑒𝑢𝐾0𝑢2+𝑌2𝑑𝑢,(5.17) where 𝐾0 is the modified Bessel function of zeroth order [13, Section 9.6.], 𝑋 and 𝑌 are spatial dimensionless coordinates, and 𝒯is a characteristic temperature,𝑣𝜁=𝑑,𝑞2𝑘𝒯=2𝜋𝑘0𝜁,𝑌=𝜁𝑦,𝑋=𝜁𝑥,Δ=𝜁𝛿.(5.18) According to what we have seen in (5.15), the maximum temperature is reached on the surface at the stationary regime. Thus, we have to analyze the maximum of the function given in (5.17) taking 𝑌=0, that is,𝑇cont(𝑋,0)=𝒯Δ𝑋𝑋𝑒𝑢𝐾0(|𝑢|)𝑑𝑢.(5.19) In order to determine the location of the maximum on the surface, firstly let us calculate the points 𝑋𝑚 where 𝑇cont(𝑋,0) has a null derivative (extrema points), 𝑑𝑇cont(𝑋,0)||||𝑑𝑋𝑋=𝑋𝑚𝑒=𝒯𝑋𝑚𝐾0||𝑋𝑚||𝑒Δ𝑋𝑚𝐾0||Δ𝑋𝑚||=0.(5.20) Therefore, 𝑋𝑚 satisfies𝑔𝑋𝑚=𝑒Δ,(5.21) where𝐾𝑔(𝑋)=0||𝑋||𝐾0||||Δ𝑋.(5.22) When 𝑣𝑑>0, the workpiece moves as indicates Figure 1, so that, now on we will consider Δ>0. Since the incoming heat flux into the workpiece is a positive magnitude, 𝑞>0, we have, 𝒯>0. Moreover, since 𝐾0 is positive for positive arguments [13, Section 9.6.], the integrand of (5.21) is also positive, thus,𝑇cont(𝑋,0)>0.(5.23)

Location of the Extrema
Assume first that 𝑋𝑚>Δ>0, so that (5.19) results in 𝐾0𝑋𝑚𝐾0𝑋𝑚Δ=𝑒Δ.(5.24) We may rewrite (5.24) as 1(Δ)=1(0), where 1(Δ)=𝑒Δ𝐾0(𝑋𝑚Δ). Since 𝐾0 is positive for positive arguments [13, Section 9.6.], we have for all 𝑋𝑚>Δ>0, 1(Δ)=𝑒Δ𝐾1𝑋𝑚Δ+𝐾0𝑋𝑚Δ>0,(5.25) that is, 1(Δ)>1(0), for 𝑋𝑚>Δ>0. Therefore, we conclude 𝑋𝑚(Δ,).(5.26)
Assume now that 𝑋𝑚<0, so that (5.21) becomes 𝐾0𝑋𝑚𝐾0Δ𝑋𝑚=𝑒Δ.(5.27) Performing the change of variables 𝑍=𝑋𝑚>0, (5.27) is equivalent to 2(Δ)=2(0), where 2(Δ)=𝑒Δ𝐾0(Δ+𝑍). Due to the integral representation [13, Equation 9.6.24], 𝐾𝜈(𝑧)=0[]exp𝑧cosh𝛼cosh𝜈𝛼𝑑𝛼,(5.28) and since for all 𝛼>0, cosh𝛼>1, we have for all 𝑍,Δ>0, 2(Δ)=𝑒Δ𝐾0(Δ+𝑍)𝐾1(Δ+𝑍)<0.(5.29) So that, 2(Δ)<2(0), for 𝑍,Δ>0. That is, (5.27) is not satisfied for 𝑋𝑚<0, 𝑋𝑚(,0).(5.30)
Finally, assume that 𝑋(0,Δ), so that |𝑋|=𝑋and |Δ𝑋|=Δ𝑋, and therefore, according to (5.22), 𝐾𝑔(𝑋)=0(𝑋)𝐾0(Δ𝑋),𝑋(0,Δ).(5.31) Since 𝑔(𝑋) is a continuous function in 𝑋(0,Δ) and, lim𝑋0𝑔(𝑋)=+,lim𝑋Δ𝑔(𝑋)=0,(5.32) according to Bolzano’s theorem, 𝑋𝑚𝑋(0,Δ)sothat𝑔𝑚=𝑒Δ.(5.33)

Uniqueness of the Extremum and Identification as Maximum
Since 𝐾0 is a positive and monotonically decreasing function for positive arguments, 𝐾0(𝑥)>0, 𝐾0(𝑥)<0 for 𝑥>0, [13, Section 9.6.] we have that 𝑔(𝑋) is a monotonically decreasing function for 𝑋(0,Δ), 𝑔𝐾(𝑋)=0(𝑋)𝐾0(Δ𝑋)+𝐾0(𝑋)𝐾0(Δ𝑋)𝐾20(Δ𝑋)<0.(5.34) Therefore, according to (5.33), !𝑋𝑚𝑋(0,Δ)sothat𝑔𝑚=𝑒Δ.(5.35) On the one hand, according to (5.26), (5.30), and (5.35), 𝑇cont(𝑋,0) has a unique extremum in 𝑋𝑚 and this one always occurs within the interval 𝑋𝑚(0,Δ). On the other hand, from (5.19) we can see that lim𝑋±𝑇cont(𝑋,0)=lim𝑋±𝒯Δ𝑋𝑋𝑒𝑢𝐾0(|𝑢|)𝑑𝑢=0.(5.36) Since 𝑇cont(𝑋,0) is a positive (5.23), continuous and differentiable function, which satisfies (5.36), the only possibility is that the extremum 𝑋𝑚 corresponds to a global maximum. Therefore, just compute a root 𝑋𝑚 of (5.21) within the interval 𝑋(0,Δ), that is taking (5.31), in order to get the location on the surface of the maximum temperature, 𝑥max=𝑋𝑚𝜁.(5.37) There is an equivalent, but more elaborated proof, in [14].

5.4. Relaxation Time

In order to estimate how rapid the transient regime is, according to (5.4), we will be close to it when, for a certain time 𝑡,𝜕𝑇cont𝑡,𝑥,𝑦𝜕𝑡=𝜖0(5.38) is satisfied. Notice that (5.38) depends on the workpiece point (𝑥,𝑦) chosen for the evaluation of 𝑡. We can define the relaxation time 𝑡, as the time that satisfies (5.38) over the maximum temperature point. According to (5.15), that point must be on the surface in the stationary state, (𝑥,𝑦)=(𝑥max,0), thus,𝜕𝑇cont𝑡,𝑥max,0=𝑞𝜕𝑡𝑘2𝑘0𝑥ERFmax,𝑡𝜋𝑡=𝜖.(5.39) Equation (5.39) can be solved numerically. In order to solve it approximately, we can expand the following function up to the first order, near the stationary regime 𝑡, [13, Equation 7.1.6]: 𝑣(𝑡,𝑧)=erf𝑑2𝑘𝑧𝑡+2𝑘𝑡𝑡𝑣erf𝑑2𝑘𝑡+𝑧𝑣𝜋𝑘𝑡exp2𝑑𝑡4𝑘.(5.40) Therefore,ERF(𝑥,𝑡)=(𝑡,𝑥)(𝑡,𝑥𝛿)𝑡𝛿𝑣𝜋𝑘𝑡exp2𝑑𝑡4𝑘.(5.41) Substituting (5.41) in (5.39), we have the following approximated equation:𝑞𝛿2𝜋𝑘0𝑡𝑣exp2𝑑𝑡4𝑘𝜖.(5.42) Using the Lambert function 𝑊 [15], we can derive the relaxation time from (5.42), arriving at,𝑡4𝑘𝑣2𝑑𝑊𝑞𝛿𝑣2𝑑8𝜋𝑘𝑘0𝜖.(5.43) Notice that in (5.43), the relaxation time is independent of the localization of the maximum on the surface 𝑥max, thus it can be computed much more rapidly.

6. Numerical Analysis

For the plots presented in this section, we have taken as grinding parameters: 𝛿=2.663×103m,𝑣𝑑=0.53 m/s, and 𝑞=5.89×107W/m2. We have considered as well a VT20 titanium alloy workpiece, whose thermal properties are 𝑘0=13W/(mK) and 𝑘=4.23×106m2/s [16]. Following the procedure described in Section 5.3, the maximum temperature in continuous grinding and its location on the workpiece surface is𝑇max=742.23K,𝑥max=7.1568×103𝛿.(6.1) In order to evaluate the relaxation time, according to (5.38), we have taken a very small parameter 𝜖=106 K/s. Taking into account (6.1), we may solve numerically (5.39) and compute the approximation given in (5.43), obtaining𝑡=8.3013×103s,𝑡1.6727×103𝑠.(6.2) Notice that the results given in (6.2) coincide in order of magnitude.

For the case of intermittent grinding we have taken in (4.1), (4.3), (4.4), and (4.7), the following wheel parameters: 𝜒=0.7𝛿 and 𝜒1=0.5𝛿, and a wheel velocity over the workpiece surface 𝑣𝑚=𝛿/𝑡. According to this data, the point period of the quasistationary regime is 𝜏=0.7𝑡, and the global period is 𝜏=2.1𝑡. Figures 7, 8, and 9 show the time evolution of the workpiece surface temperature for 𝑡(0,𝑡), for continuous and intermittent up- and downgrinding, respectively. As can be seen, the temporal evolution of up and down grinding is quite different from each other, but in both cases, the continuous profile is a limit boundary. Figure 10 compares the temperature time evolution in 𝑥max of continuous grinding with intermittent up- and downgrinding. We may highlight that the relaxation time obtained for the continuous case is a good estimation for the transient regime in the intermittent case. We may notice also how upgrinding nearly saturates the maximum temperature of the continuous case, but this does not occur in downgrinding. We may evaluate numerically the maximum temperature, both intermittent up- and downgrindings,𝑇downmax=741.41K,𝑇upmax=591.29K.(6.3)

927876.fig.007
Figure 7: Surface temperature evolution in continuous grinding 𝑇cont(𝑡,𝑥,0) for 𝑥(𝛿,𝛿) and 𝑡=𝑡𝑚/5, taking 𝑚=1,2,3,4,5 (red, orange, green, blue, magenta, resp.).
927876.fig.008
Figure 8: Surface temperature evolution in intermittent downgrinding 𝑇(𝑡,𝑥,0) for 𝑥(𝛿,𝛿) and 𝑡=𝑡𝑚/5, taking 𝑚=1,2,3,4,5 (red, orange, green, blue, magenta, resp.).
927876.fig.009
Figure 9: Surface temperature evolution in intermittent upgrinding 𝑇(𝑡,𝑥,0) for 𝑥(𝛿,𝛿) and 𝑡=𝑡𝑚/5, taking 𝑚=1,2,3,4,5 (red, orange, green, blue, magenta, resp.).
927876.fig.0010
Figure 10: Time evolution of 𝑇cont(𝑡,𝑥max,0) and 𝑇(𝑡,𝑥max,0), for 𝑡(0,2𝑡).

Figure 11 shows the time evolution of the quasistationary regime on the surface for a friction period 𝜏. For 𝑥<0, the temperature oscillates as a wave. This is because the heat flux pulses produced at the contact zone are propagated along the surface just ground. Figure 12 shows the time evolution of the temperature in 𝑥max for 𝑡(0,2𝜏). On the one hand, we may check that the quasistationary regime has a period 𝜏, as it was commented in (5.9). On the other hand, we may notice that the quasistationary regime is reached when the temperature in the continuous case is saturated. Therefore, the relaxation time 𝑡 defined for the continuous case is a good measurement for the transient regime when we have a quasistationary regime in intermittent grinding. In Figures 14 and 15, we have plotted the temperature fields at 𝑡=𝑡, in the cases of up- and downgrinding, respectively. We may realize that both temperature fields are quite different from each other. Figure 13 shows the field temperature for the continuous case. If we compare the temperature field in the continuous case with the intermittent one (up- or downgrinding), we may observe that an intermittent friction distorts the temperature field producing thermic waves inside the workpiece.

927876.fig.0011
Figure 11: Time evolution of 𝑇(𝑡,𝑥,𝑦) for 𝑥(𝛿,𝛿) and 𝑡=𝑚𝜏/5, taking 𝑚=1,2,3,4,5 (red, orange, green, blue, magenta, resp.).
927876.fig.0012
Figure 12: Comparison of the time evolution on 𝑥max for 𝑡(0,2𝜏).
927876.fig.0013
Figure 13: Field temperature 𝑇cont(𝑡,𝑥,𝑦) for (𝑥,𝑦)(𝛿,𝛿)×(0,𝛿/10). Contours give the temperature in K.
927876.fig.0014
Figure 14: Field temperature 𝑇(𝑡,𝑥,𝑦) for (𝑥,𝑦)(𝛿,𝛿)×(0,𝛿/10), in the case of downgrinding. Contours give the temperature in K.
927876.fig.0015
Figure 15: Field temperature 𝑇(𝑡,𝑥,𝑦) for (𝑥,𝑦)(𝛿,𝛿)×(0,𝛿/10) in the case of upgrinding. Contours give the temperature in K.

7. Conclusions

We have derived a closed analytical solution for the time evolution of the temperature field in dry grinding for any time-dependent friction function. Our result is based on the Samara-Valencia model [9], solving explicitly the evolution of temperature field for the case of dry grinding. We find this solution solving a recurrence equation by successive approximations. We have proved that this solution is unique. An analytical solution of this type has the advantage to be straightforwardly computable, plotting the graphs very rapidly. Also, the dependence of the grinding parameters on the temperature field can be studied. The latter is quite useful for the engineering optimization of the grinding process.

We apply our solution to continuous and intermittent up- and downgrinding. We have tested numerically that the time evolution of up- and downgrinding is quite different from each other. In continuous grinding, we have proved that the maximum temperature occurs at the stationary regime within the friction zone on the surface. In order to graph the evolution of the temperature field, we have obtained a useful approximation for the characteristic time of the transient regime. Comparing the plots of continuous and intermittent grinding for the same workpiece and grinding parameters, we conclude that the behavior of the intermittent case is more complicated in detail, but in general the magnitude of temperature field is lower. The latter is quite understandable because, in intermittent grinding, the amount of energy per unit time entering into the workpiece due to friction is less than in the continuous case. Therefore, the temperature plot for the continuous grinding acts as a boundary for the intermittent case.

Also, we have tested numerically that the relaxation time obtained for continuous grinding is a good estimation for the characteristic time of the transient regime in the intermittent case. Finally, we have obtained an expression for the quasistationary regime in intermittent grinding, in which the field temperature oscillates periodically.

Acknowledgments

The authors wish to thank the financial support received from Generalitat Valenciana under Grant GVA 3012/2009 and from Universidad Politécnica de Valencia under Grant PAID-06-09.

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