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

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 [1–5] 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 𝑣_𝑑 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)=𝑘𝜕_𝑥𝑥𝑇(𝑡,𝑥,𝑦)+𝜕_𝑦𝑦𝑇(𝑡,𝑥,𝑦)−𝑣_𝑑𝜕_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,𝑦≥0. 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⁾(𝑡,𝑥,𝑦)∶=−14𝜋𝑘₀^𝑡0𝑑𝑠𝑠exp−4𝑘𝑦𝑠²^∞−∞𝑑𝑥^𝑑𝑡−𝑠,𝑥^exp−𝑥^−𝑥−𝑣_𝑑𝑠²𝑇4𝑘𝑠,(2.5)₍₁₎(𝑡,𝑥,𝑦)∶=14𝜋^𝑡0𝑑𝑠𝑠exp−4𝑘𝑦𝑠²^∞−∞𝑑𝑥^2𝑦𝑘𝑠−𝑏𝑡−𝑠,𝑥^×𝑇𝑡−𝑠,𝑥_𝑘₀,0exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠.(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).

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⁾(𝑡,𝑥,𝑦)=𝑦8𝜋𝑘^𝑡0exp−𝑦²4𝑘𝑠𝑑𝑠²𝑠×^∞−∞𝑑𝑥^𝑇𝑡−𝑠,𝑥^,0exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠.(3.2) In order to tackle the integral equation given in (3.2), let us define the following integral operators𝑇⁽¹⁾(𝑡,𝑥,𝑦)=ℶ_𝑦[𝑇(𝑡,𝑥,0)],(3.3) whereℶ_𝑦[𝑇(𝑡,𝑥,0)]∶=𝑦8𝜋𝑘^𝑡0𝑑𝑠𝑠²exp−𝑦²4𝑘𝑠ℵ^𝑠ℵ[𝑇(𝑡,𝑥,0)],(3.4)_𝑠[𝑇(𝑡,𝑥,0)]∶=^∞−∞𝑑𝑥^𝑇𝑡−𝑠,𝑥^,0exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠.(3.5) Therefore, taking into account (3.3), we may rewrite (2.4) as𝑇(𝑡,𝑥,𝑦)=𝑇⁽⁰⁾(𝑡,𝑥,𝑦)+ℶ_𝑦[𝑇(𝑡,𝑥,0)].(3.6)

3.2. The ℵ_𝑠 Operator

Let us calculate the ℵ_𝑠 operator over the frictional term 𝑇⁽⁰⁾ of the temperature field. According to (2.5), 𝑇⁽⁰⁾ may be expressed as𝑇⁽⁰⁾𝑡−𝑠,𝑥^,0=−14𝜋𝑘₀^0𝑡−𝑠𝑑𝜎𝜎^∞−∞𝑑𝜉𝑑(𝑡−𝑠−𝜎,𝜉)exp−𝜉−𝑥^−𝑣_𝑑𝜎²4𝑘𝜎.(3.7) Therefore, substituting (3.7) in (3.5), and reordering the integrals by Fubini’s theorem, we obtainℵ_𝑠𝑇⁽⁰⁾(𝑡,𝑥,0)=−14𝜋𝑘₀^∞−∞𝑑𝜉^0𝑡−𝑠𝑑𝜎×𝜎𝑑(𝑡−𝑠−𝜎,𝜉)_∞−∞𝑑𝑥^exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠−𝜉−𝑥^−𝑣_𝑑𝜎²4𝑘𝜎.(3.8) Expanding the exponent of the integrand given in (3.8), we arrive atℵ_𝑠𝑇⁽⁰⁾(𝑡,𝑥,0)=−14𝜋𝑘₀exp−𝑥+𝑣^𝑑𝑠²×4𝑘𝑠_∞−∞𝑑𝜉exp𝑣^𝑑𝜉×2𝑘_0𝑡−𝑠𝑑𝜎𝜎𝑑(𝑡−𝑠−𝜎,𝜉)exp−4𝜉𝑘²𝜎−𝑣^2𝑑𝜎×4𝑘_∞−∞𝑑𝑥^exp−𝑥^24𝑘𝑠𝑠+𝜎𝜎+2𝑥𝑘^𝑥𝜎𝑠+𝜎𝜉𝑠.(3.9) The last integral given in (3.9) can be calculated [10, Equation 3.323.2], so that,ℵ_𝑠𝑇⁽⁰⁾(𝑡,𝑥,0)=−√𝑘𝑠2√𝜋𝑘₀exp−𝑥+𝑣^𝑑𝑠²×4𝑘𝑠_∞−∞𝑑𝜉exp𝑣^𝑑𝜉×2𝑘_0𝑡−𝑠𝑑𝜎𝑑(𝑡−𝑠−𝜎,𝜉)√𝜎√𝑠+𝜎exp4(𝑘𝑥𝑠𝜎𝜎+(𝑠𝜉+𝑠)𝜎²)−4𝜉𝑘²𝜎−𝑣^2𝑑𝜎4𝑘.(3.10) Once again, expanding the exponent of the last integrand given in (3.10) and simplifying, we arrive atℵ_𝑠𝑇⁽⁰⁾(𝑡,𝑥,0)=−√𝑘𝑠2√𝜋𝑘₀exp−𝑥𝑣^𝑑2𝑘^∞−∞𝑑𝜉exp𝑣^𝑑𝜉×2𝑘_0𝑡−𝑠𝑑𝜎𝑑(𝑡−𝑠−𝜎,𝜉)√𝜎√𝑠+𝜎exp−4(𝑘𝑥(−𝑠+𝜉)𝜎²)−4𝑣𝑘^2𝑑(𝜎+𝑠).(3.11) Let us define𝐼_𝜎∶=^0𝑡−𝑠𝑑𝜎𝑑(𝑡−𝑠−𝜎,𝜉)√𝜎√𝑠+𝜎exp−4(𝑘𝑥(−𝑠+𝜉)𝜎²)−4𝑣𝑘^2𝑑(𝜎+𝑠).(3.12) We can calculate (3.12) performing the substitution, 𝜇=𝜎+𝑠, and introducing the Heaviside function 𝐻(𝑥), so that,𝐼_𝜎=^𝑡0𝑑𝜇𝑑(𝑡−𝜇,𝜉)√𝜇−𝑠√𝜇exp−(𝑥4−𝑘𝜇𝜉)²−4𝑣𝑘^2𝑑𝜇𝐻(𝜇−𝑠).(3.13) Substituting (3.13) in (3.11) and simplifying, we getℵ_𝑠𝑇⁽⁰⁾(𝑡,𝑥,0)=−√𝑘𝑠2√𝜋𝑘₀^∞−∞𝑑𝜉^𝑡0𝑑𝜇𝑑(𝑡−𝜇,𝜉)√𝜇−𝑠√𝜇exp−𝜉−𝑥−𝑣^𝑑𝜇²4𝑘𝜇𝐻(𝜇−𝑠).(3.14)

3.3. The ℶ_𝑦 Operator

Substituting the expression obtained in (3.14) into (3.4) and reordering the integrals, we haveℶ_𝑦𝑇⁽⁰⁾(𝑡,𝑥,0)=−𝑦16𝜋^3/2𝑘₀√𝑘^∞−∞×𝑑𝜉_𝑡0𝑑𝜇𝑑(𝑡−𝜇,𝜉)√𝜇exp−𝜉−𝑥−𝑣^𝑑𝜇²×4𝑘𝜇_𝑡0𝑑𝑠𝐻(𝜇−𝑠)𝑠^3/2√𝜇−𝑠exp−4𝑘𝑦𝑠².(3.15) Let us define𝐼_𝑠∶=^𝑡0𝑑𝑠𝐻(𝜇−𝑠)𝑠^3/2√𝜇−𝑠exp−4𝑘𝑦𝑠².(3.16) Since 𝜇∈[0,𝑡], the integral given in (3.16) can be expressed in the following way:𝐼_𝑠=^𝜇0𝑑𝑠𝑠^3/2√𝜇−𝑠exp−4𝑘𝑦𝑠².(3.17) In order to calculate (3.17), we can perform the following substitutions: 𝑠=𝜇/𝑡, 𝑤=√𝑡−1 and 𝑟=𝑦𝑤/2√𝑘𝜇, leading to𝐼_𝑠=2√𝜋𝑘𝑦√𝜇exp−4𝑦𝑘²𝜇.(3.18) Therefore, substituting (3.18) in (3.15) and changing the integration order, we arrive at ℶ_𝑦𝑇⁽⁰⁾(𝑡,𝑥,0)=−18𝜋𝑘₀^𝑡0𝑑𝜇𝜇exp−4𝑦𝑘²𝜇^∞−∞𝑑𝜉𝑑(𝑡−𝜇,𝜉)exp−𝑥−𝜉−𝑣^𝑑𝜇²4𝑘𝜇.(3.19) Remembering the expression for 𝑇⁽⁰⁾ given in (2.5), we concludeℶ_𝑦𝑇⁽⁰⁾(𝑡,𝑥,0)=12𝑇⁽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)=ℶ_𝑦𝑇₀(𝑡,𝑥,0)=ℶ_𝑦𝑇⁽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)+ℶ_𝑦𝑇₀(𝑡,𝑥,0).(3.24) In general, the 𝑛th (𝑛=0,1,2,…) approximation is𝑇_𝑛(𝑡,𝑥,𝑦)∶=𝑇⁽⁰⁾(𝑡,𝑥,𝑦)+ℶ_𝑦𝑇_𝑛−1(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 as𝑇₁(𝑡,𝑥,𝑦)=32𝑇⁽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 ℶ_𝑦 is linear, we obtain𝑇₂(𝑡,𝑥,𝑦)==𝑇𝑇⁽⁰⁾₍₀₎(𝑡,𝑥,𝑦)+ℶ_𝑦𝑇₁(𝑡,𝑥,0)(𝑡,𝑥,𝑦)+32ℶ^𝑦𝑇⁽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:𝑇_𝑛(𝑡,𝑥,𝑦)=2^𝑛+1−12^𝑛𝑇⁽⁰⁾(𝑡,𝑥,𝑦),(3.29) that can be proved by induction, 𝑇_𝑛+1(𝑡,𝑥,𝑦)==𝑇𝑇⁽⁰⁾₍₀₎(𝑡,𝑥,𝑦)+ℶ_𝑦𝑇_𝑛(𝑡,𝑥,0)(𝑡,𝑥,𝑦)+2^𝑛+1−12^𝑛=2ℶ_𝑦𝑇⁽⁰⁾(𝑡,𝑥,0)_𝑛+2−12^𝑛+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)=lim_𝑛→∞𝑇_𝑛(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)==𝑇𝑇⁽⁰⁾₍₀₎(𝑡,𝑥,𝑦)+ℶ_𝑦[𝑇(𝑡,𝑥,0)]=2𝑇(₍₀₎𝑡,𝑥,𝑦)+2ℶ_𝑦𝑇⁽⁰⁾(𝑡,𝑥,0)(𝑡,𝑥,𝑦).(3.32) Taking into account (2.5), we conclude that the time evolution of the temperature field may be expressed as𝑇(𝑡,𝑥,𝑦)=−12𝜋𝑘₀^𝑡0𝑑𝑠𝑠exp−4𝑘𝑦𝑠²^∞−∞𝑑𝑥^𝑑𝑡−𝑠,𝑥^exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠.(3.33)

3.5. Uniqueness of the Solution

3.5.1. Bound Limit for ℶ₀

To prove the uniqueness of the solution of the integral equation (3.6), let us calculate first the value of the ℶ_𝑦 operator over a constant. According to (3.5), we haveℵ_𝑠[1]=^∞−∞exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠𝑑𝑥^.(3.34) Performing the substitution: 𝑢=(𝑥^−𝑥−𝑣_𝑑𝑠)/2√𝑘𝑠, (3.33) results inℵ_𝑠[1]=2√𝑘𝑠^∞−∞𝑒^−𝑢2𝑑𝑢=2√𝜋𝑘𝑠.(3.35) Applying (3.35) to (3.4), we have ℶ_𝑦[1]=𝑦4√𝜋𝑘^𝑡0𝑑𝑠𝑠^3/2exp−𝑦²4𝑘𝑠.(3.36) Performing the substitution: 𝑢=𝑦/2√𝑘𝑠, we haveℶ_𝑦[1]=1√𝜋^{∞√𝑦/2𝑘𝑡}𝑒^−𝑢2𝑑𝑢=12erfc2√𝑦𝑘𝑡.(3.37) Therefore,ℶ₀[1]=12.(3.38)

Let us consider now a function 𝑓(𝑡,𝑥,𝑦) whose maximum value taking 𝑦=0 is 𝑓_max, that is,𝑓(𝑡,𝑥,0)≤𝑓_max.(3.39) Applying ℶ₀ to (3.39) and taking into account that ℶ₀ is a linear operator,ℶ₀[𝑓(𝑡,𝑥,0)]≤ℶ₀𝑓_max=𝑓_maxℶ₀[1].(3.40) Thus, according to (3.38),ℶ₀[𝑓(𝑡,𝑥,0)]≤𝑓^max2.(3.41) Note that, if we apply ℶ₀ to (3.39) and we take into account (3.37), we have ℶ⁰⁽²⁾[𝑓(𝑡,𝑥,0)]≤ℶ₀𝑓2^max=𝑓2^max2.(3.42) So, in general, for all 𝑛∈ℕ,ℶ^0(𝑛)[𝑓(𝑡,𝑥,0)]≤𝑓^max2^𝑛.(3.43)

3.5.2. Resolution of the Uniqueness

If 𝑇_𝐴(𝑡,𝑥,𝑦) and 𝑇_𝐵(𝑡,𝑥,𝑦) are solutions of (3.6), we have𝑇𝑇_𝐴(𝑡,𝑥,𝑦)=𝑇⁽⁰⁾(𝑡,𝑥,𝑦)+ℶ_𝑦𝑇_𝐴(𝑡,𝑥,0),(3.44)_𝐵(𝑡,𝑥,𝑦)=𝑇⁽⁰⁾(𝑡,𝑥,𝑦)+ℶ_𝑦𝑇_𝐵(𝑡,𝑥,0).(3.45) Subtracting (3.45) from (3.44) and taking into account that ℶ_𝑦 is a linear operator,𝑇_𝐴(𝑡,𝑥,𝑦)−𝑇_𝐵(𝑡,𝑥,𝑦)=ℶ_𝑦𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0).(3.46) Taking 𝑦=0 in (3.46),𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0)=ℶ₀𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0).(3.47) Recursive substitution of (3.47) yields𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0)=ℶ^0(𝑛)𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0).(3.48) If we take in (3.43) as a function 𝑓, 𝑓₁(𝑡,𝑥,0)=𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0),(3.49) according to (3.48), we have that, for all 𝑛∈ℕ,𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0)≤𝑓^1,max2^𝑛,(3.50) where 𝑓_1,max is the maximum value of 𝑓₁(𝑡,𝑥,0). Taking the limit in (3.50), 𝑇_𝐴(𝑡,𝑥,0)−𝑇_𝐵(𝑡,𝑥,0)≤lim_𝑛→∞𝑓_1,max2^𝑛=0,(3.51) so that,𝑇_𝐴(𝑡,𝑥,0)≤𝑇_𝐵(𝑡,𝑥,0).(3.52) Note that in (3.48) we can exchange labels 𝐴 and 𝐵, 𝑇_𝐵(𝑡,𝑥,0)−𝑇_𝐴(𝑡,𝑥,0)=ℶ^0(𝑛)𝑇_𝐵(𝑡,𝑥,0)−𝑇_𝐴(𝑡,𝑥,0).(3.53) Thus, taking now the function 𝑓₂(𝑡,𝑥,0)=𝑇_𝐵(𝑡,𝑥,0)−𝑇_𝐴(𝑡,𝑥,0),(3.54) we obtain that 𝑇_𝐵(𝑡,𝑥,0)−𝑇_𝐴(𝑡,𝑥,0)≤lim_𝑛→∞𝑓_2,max2^𝑛=0,(3.55) that is,𝑇_𝐵(𝑡,𝑥,0)≤𝑇_𝐴(𝑡,𝑥,0).(3.56) From (3.52) and (3.56), we conclude that both solutions on the surface are equal,𝑇_𝐴(𝑡,𝑥,0)=𝑇_𝐵(𝑡,𝑥,0).(3.57) Applying ℶ_𝑦 to (3.57),ℶ_𝑦𝑇_𝐴(𝑡,𝑥,0)=ℶ_𝑦𝑇_𝐵(𝑡,𝑥,0),(3.58) and substituting (3.58) in (3.44), we have that𝑇_𝐴(𝑡,𝑥,𝑦)=𝑇⁽⁰⁾(𝑡,𝑥,𝑦)+ℶ_𝑦𝑇_𝐵(𝑡,𝑥,0).(3.59) Comparing (3.45) with (3.59), we finally obtain 𝑇_𝐴(𝑡,𝑥,𝑦)=𝑇_𝐵(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).

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 𝑣_𝑚=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∈(𝑎_𝑗,𝑏_𝑗), 𝑗=0,…,𝑛+1, where 𝑗 indicates a wheel tooth. Notice that there can be up to 𝑛+2 teeth within the contact zone, where𝑛∶=𝛿𝜒.(4.1) Note, also, that the friction limits are time dependent: 𝑎_𝑗=𝑎_𝑗(𝑡) and 𝑏_𝑗=𝑏_𝑗(𝑡). If the incoming heat flux 𝑞 is constant for every point where friction occurs, we may write the friction function as𝑑(𝑡,𝑥)=−𝑞^𝑛+1_𝑗=0𝐻𝑥−𝑎_𝑗(𝑡)𝐻𝑏_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, 𝑎_𝑗 and 𝑏_𝑗, let us define the spatial period,𝜒∶=(𝑛+2)𝜒.(4.3) According to Figure 5, the 𝑔_𝑗(𝑡) points, 𝑗=0,…,𝑛+1, are initially over the period 𝜒,𝑔_𝑗(𝑡)∶=(−1)^𝜙𝑣_𝑚𝑡+𝜒−𝑗𝜒,(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 function𝑓_𝑗(𝑡)=𝑔_𝑗(𝑡)−𝑔^𝑗(𝑡)𝜒𝜒.(4.5) We want as well that 𝑏_𝑗∈[0,𝛿], thus,𝑏_𝑗(𝑡)=minmax𝑓_𝑗(𝑡),0,𝛿=⎧⎪⎨⎪⎩𝑏^𝑗0(𝑡)0𝑏_𝑗<𝑏_𝑗(𝑡)<𝛿,𝛿𝑏_𝑗(𝑡)≤0,(𝑡)≥𝛿.(4.6) Similarly, since the tooth width is 𝜒₁,𝑎_𝑗(𝑡)=minmax𝑓_𝑗(𝑡)−𝜒₁,0,𝛿.(4.7) The min and max functions are given by min(𝑎,𝑏)=𝑎+𝑏−||𝑎−𝑏||max(𝑎,𝑏)=𝑎+𝑏+2||𝑎−𝑏||,2.(4.8)

Figure 3 

Friction function 𝑑(𝑡,𝑥) for time 𝑡1 highlighted in red.

Figure 4 

Friction function 𝑑(𝑡,𝑥) for time 𝑡2>𝑡1 highlighted in red.

Figure 5 

Initial location of the 𝑔𝑗(𝑡) points, for 𝑛=2.

Figure 6 

Upgrinding and downgrinding according to the wheel rotation.

4.2. Temperature Field

Substituting (4.2) in (3.33), we obtain𝑇(𝑡,𝑥,𝑦)=𝑞2𝜋𝑘₀^𝑡0𝑑𝑠×𝑠exp−4𝑘𝑦𝑠²_𝑛+1_𝑗=0^{𝑏𝑗𝑎(𝑡−𝑠)}_{𝑗(𝑡−𝑠)}𝑑𝑥^exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠.(4.9) Let us evaluate the integral over the 𝑥^ variable in (4.9),𝐼_𝑥′∶=^𝑛+1_𝑗=0^{𝑏𝑗𝑎(𝑡−𝑠)}_{𝑗(𝑡−𝑠)}exp−𝑥^−𝑥−𝑣_𝑑𝑠²4𝑘𝑠𝑑𝑥^.(4.10) Performing the substitution,𝑢=𝑥^−𝑥−𝑣_𝑑𝑠2√𝑘𝑠,(4.11) and taking into account the properties of the error function, we get𝐼_𝑥′=√𝜋𝑘𝑠ERF(𝑡,𝑥,𝑠),(4.12) where we have defined the functionERF(𝑡,𝑥,𝑠)∶=^𝑛+1_𝑗=0erf𝑥+𝑣^𝑑𝑠−𝑎_𝑗(𝑡−𝑠)2√𝑘𝑠−erf𝑣^𝑑𝑠+𝑥−𝑏_𝑗(𝑡−𝑠)2√𝑘𝑠.(4.13) Substituting (4.12) in (4.9), we obtain the following expression for the temperature field: 𝑇(𝑡,𝑥,𝑦)=𝑞√𝑘2√𝜋𝑘₀^𝑡0𝑑𝑠√𝑠exp−4𝑘𝑦𝑠²ERF(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, 𝑎𝑏₀(𝑡)=0,₀(𝑡)=𝛿.(5.1) Therefore, we can redefine (4.12) as ERF_cont(𝑥,𝑠)∶=erf𝑥+𝑣^𝑑𝑠2√𝑘𝑠−erf𝑣^𝑑𝑠+𝑥−𝛿2√𝑘𝑠,(5.2) obtaining, according to (4.13), the following temperature field:𝑇_cont(𝑡,𝑥,𝑦)=𝑞√𝑘2√𝜋𝑘₀^𝑡0𝑑𝑠√𝑠exp−4𝑘𝑦𝑠²ERF^cont(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𝜕𝑇_cont(𝑡,𝑥,𝑦)𝜕𝑡=𝑞√𝑘2𝑘₀ERF_cont(𝑥,𝑡)√𝜋𝑡exp−4𝑘𝑦𝑡².(5.5) Taking the limit of (5.4), knowing that erf(±∞)=±1, we can check that the stationary regime is reached when 𝑡→∞:lim_𝑡→∞ERF_cont(𝑥,𝑡)=lim_𝑡→∞erf𝑥+𝑣^𝑑𝑡2√𝑘𝑡−erf𝑣^𝑑𝑡+𝑥−𝛿2√𝑘𝑡=0,(5.6) so that,lim_𝑡→∞𝜕𝑇_cont(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 𝑎_𝑗(𝑡) and 𝑏_𝑗(𝑡) are periodic functions (4.6)-(4.7), according to (4.14), we may define the quasistationary regime as𝑇_∞(𝑡,𝑥,𝑦)∶=𝑞√𝑘2√𝜋𝑘₀^∞0𝑑𝑠√𝑠exp−4𝑘𝑦𝑠²ERF(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 is𝜏∶=𝜒𝑣_𝑚.(5.9) However, the global consideration of the 𝑑(𝑡,𝑥) plot indicates the following temporal period:𝜏∶=𝜒𝑣_𝑚.(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 haveERF_cont(𝑥,𝑡)>0𝑡>0,𝑥∈ℝ.(5.11) Therefore, the temperature on a given point (𝑥,𝑦) of the workpiece is a monotonically increasing function,𝜕𝑇_cont(𝑡,𝑥,𝑦)𝜕𝑡>0𝑦,𝑡>0,𝑥∈ℝ.(5.12) Equation (5.12) means that the maximum temperature must be reached in the stationary state, 𝑡→∞. Moreover, as in (5.11), we haveERF_cont(𝑥,𝑠)>0𝑠∈(0,𝑡),𝑥∈ℝ,(5.13) so that, for 𝑦>0,𝜕𝑇_cont(𝑡,𝑥,𝑦)𝜕𝑦=−4√𝑞𝜋𝑦𝑘𝑘₀^𝑡0𝑑𝑠𝑠^3/2exp−𝑦²4𝑘𝑠ERF^cont(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,𝑇_max=lim_𝑡→∞𝑇_cont𝑡,𝑥_max,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 𝑇_cont(𝑥,𝑦)∶=lim_𝑡→∞𝑇_cont(𝑡,𝑥,𝑦),(5.16) according to [12], we have,𝑇_cont(𝑋,𝑌)=𝒯^{Δ−𝑋−𝑋}𝑒^𝑢𝐾₀√𝑢²+𝑌²𝑑𝑢,(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,𝜁∶=𝑣^𝑑𝒯∶=2𝑞𝑘,𝑋Δ𝑌∶∶∶===2𝜋𝜁𝜁𝜁𝑘𝑦𝛿𝑥,₀.,𝜁,(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,𝑇_cont(𝑋,0)=𝒯^{Δ−𝑋−𝑋}𝑒^𝑢𝐾₀(|𝑢|)𝑑𝑢.(5.19) In order to determine the location of the maximum on the surface, firstly let us calculate the points 𝑋_𝑚 where 𝑇_cont(𝑋,0) has a null derivative (extrema points), 𝑑𝑇_cont(𝑋,0)𝑑𝑋||||_{𝑋=𝑋𝑚}=𝒯𝑒^−𝑋𝑚𝐾₀||𝑋_𝑚||−𝑒^{Δ−𝑋𝑚}𝐾₀||Δ−𝑋_𝑚||=0.(5.20) Therefore, 𝑋_𝑚 satisfies𝑔𝑋_𝑚=𝑒^Δ,(5.21) where𝑔(𝑋)∶=𝐾⁰||𝑋||𝐾₀||Δ−𝑋||.(5.22) When 𝑣_𝑑>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,𝑇_cont(𝑋,0)>0.(5.23)

Location of the Extrema
Assume first that 𝑋_𝑚>Δ>0, so that (5.19) results in 𝐾₀𝑋_𝑚𝐾₀𝑋_𝑚−Δ=𝑒^Δ.(5.24) We may rewrite (5.24) as ℎ₁(Δ)=ℎ₁(0), where ℎ₁(Δ)∶=𝑒^Δ𝐾₀(𝑋_𝑚−Δ). Since 𝐾₀ is positive for positive arguments [13, Section 9.6.], we have for all 𝑋_𝑚>Δ>0, ℎ^1(Δ)=𝑒^Δ𝐾₁𝑋_𝑚−Δ+𝐾₀𝑋_𝑚−Δ>0,(5.25) that is, ℎ₁(Δ)>ℎ₁(0), for 𝑋_𝑚>Δ>0. Therefore, we conclude 𝑋_𝑚∉(Δ,∞).(5.26)
Assume now that 𝑋_𝑚<0, so that (5.21) becomes 𝐾₀−𝑋_𝑚𝐾₀Δ−𝑋_𝑚=𝑒^Δ.(5.27) Performing the change of variables 𝑍=−𝑋_𝑚>0, (5.27) is equivalent to ℎ₂(Δ)=ℎ₂(0), where ℎ₂(Δ)∶=𝑒^Δ𝐾₀(Δ+𝑍). Due to the integral representation [13, Equation 9.6.24], 𝐾_𝜈(𝑧)=^∞0exp[−𝑧cosh𝛼]cosh𝜈𝛼𝑑𝛼,(5.28) and since for all 𝛼>0, cosh𝛼>1, we have for all 𝑍,Δ>0, ℎ^2(Δ)=𝑒^Δ𝐾₀(Δ+𝑍)−𝐾₁(Δ+𝑍)<0.(5.29) So that, ℎ₂(Δ)<ℎ₂(0), for 𝑍,Δ>0. That is, (5.27) is not satisfied for 𝑋_𝑚<0, 𝑋_𝑚∉(−∞,0).(5.30)
Finally, assume that 𝑋∈(0,Δ), so that |𝑋|=𝑋and |Δ−𝑋|=Δ−𝑋, and therefore, according to (5.22), 𝑔(𝑋)=𝐾⁰(𝑋)𝐾₀(Δ−𝑋),𝑋∈(0,Δ).(5.31) Since 𝑔(𝑋) is a continuous function in 𝑋∈(0,Δ) and, lliimm_{𝑋→0𝑋→Δ}𝑔(𝑋)=+∞,𝑔(𝑋)=0,(5.32) according to Bolzano’s theorem, ∃𝑋_𝑚∈(0,Δ)sothat𝑔𝑋_𝑚=𝑒^Δ.(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, 𝐾^0(𝑥)<0 for 𝑥>0, [13, Section 9.6.] we have that 𝑔(𝑋) is a monotonically decreasing function for 𝑋∈(0,Δ), 𝑔^(𝑋)=𝐾^0(𝑋)𝐾₀(Δ−𝑋)+𝐾₀(𝑋)𝐾^0(Δ−𝑋)𝐾²⁰(Δ−𝑋)<0.(5.34) Therefore, according to (5.33), ∃!𝑋_𝑚∈(0,Δ)sothat𝑔𝑋_𝑚=𝑒^Δ.(5.35) On the one hand, according to (5.26), (5.30), and (5.35), 𝑇_cont(𝑋,0) has a unique extremum in 𝑋_𝑚 and this one always occurs within the interval 𝑋_𝑚∈(0,Δ). On the other hand, from (5.19) we can see that lim_𝑋→±∞𝑇_cont(𝑋,0)=lim_𝑋→±∞𝒯^{Δ−𝑋−𝑋}𝑒^𝑢𝐾₀(|𝑢|)𝑑𝑢=0.(5.36) Since 𝑇_cont(𝑋,0) is a positive (5.23), continuous and differentiable function, which satisfies (5.36), the only possibility is that the extremum 𝑋_𝑚 corresponds to a global maximum. Therefore, just compute a root 𝑋_𝑚 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, 𝑥_max=𝑋^𝑚𝜁.(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,𝜕𝑇_cont𝑡,𝑥,𝑦𝜕𝑡=𝜖≈0(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)=(𝑥_max,0), thus,𝜕𝑇_cont𝑡^∗,𝑥_max,0𝜕𝑡=𝑞√𝑘2𝑘₀ERF𝑥_max,𝑡^∗√𝜋𝑡^∗=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]: ℎ(𝑡,𝑧)∶=erf𝑣^𝑑2√𝑘√𝑡+𝑧2√𝑘𝑡∼^𝑡→∞erf𝑣^𝑑2√𝑘√𝑡+𝑧√𝜋𝑘𝑡exp−𝑣^2𝑑𝑡4𝑘.(5.40) Therefore,ERF(𝑥,𝑡)=ℎ(𝑡,𝑥)−ℎ(𝑡,𝑥−𝛿)∼_𝑡→∞𝛿√𝜋𝑘𝑡exp−𝑣^2𝑑𝑡4𝑘.(5.41) Substituting (5.41) in (5.39), we have the following approximated equation:𝑞𝛿2𝜋𝑘₀𝑡^∗exp−𝑣^2𝑑𝑡^∗4𝑘≈𝜖.(5.42) Using the Lambert function 𝑊 [15], we can derive the relaxation time from (5.42), arriving at,𝑡^∗≈4𝑘𝑣^2𝑑𝑊𝑞𝛿𝑣^2𝑑8𝜋𝑘𝑘₀𝜖.(5.43) Notice that in (5.43), the relaxation time is independent of the localization of the maximum on the surface 𝑥_max, 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=2.663×10⁻³m,𝑣_𝑑=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=4.23×10⁻⁶m²/s [16]. Following the procedure described in Section 5.3, the maximum temperature in continuous grinding and its location on the workpiece surface is𝑇_max=742.23K,𝑥_max=7.1568×10⁻³𝛿.(6.1) In order to evaluate the relaxation time, according to (5.38), we have taken a very small parameter 𝜖=10⁻⁶ K/s. Taking into account (6.1), we may solve numerically (5.39) and compute the approximation given in (5.43), obtaining𝑡^∗=8.3013×10⁻³s,𝑡^∗≈1.6727×10⁻³𝑠.(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 𝑣_𝑚=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 𝑥_max 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,𝑇^down_max=741.41K,𝑇^up_max=591.29K.(6.3)

Figure 7 

Surface temperature evolution in continuous grinding 𝑇cont(𝑡,𝑥,0) for 𝑥∈(−𝛿,𝛿) and 𝑡=𝑡∗𝑚/5, taking 𝑚=1,2,3,4,5 (red, orange, green, blue, magenta, resp.).

Figure 8 

Surface temperature evolution in intermittent downgrinding 𝑇(𝑡,𝑥,0) for 𝑥∈(−𝛿,𝛿) and 𝑡=𝑡∗𝑚/5, taking 𝑚=1,2,3,4,5 (red, orange, green, blue, magenta, resp.).

Figure 9 

Surface temperature evolution in intermittent upgrinding 𝑇(𝑡,𝑥,0) for 𝑥∈(−𝛿,𝛿) and 𝑡=𝑡∗𝑚/5, taking 𝑚=1,2,3,4,5 (red, orange, green, blue, magenta, resp.).

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 𝑥_max 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.

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.).

Figure 12 

Comparison of the time evolution on 𝑥max for 𝑡∈(0,2𝜏).

Figure 13 

Field temperature 𝑇cont(𝑡∗,𝑥,𝑦) for (𝑥,𝑦)∈(−𝛿,𝛿)×(0,𝛿/10). Contours give the temperature in K.

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.

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.