Experimental Heat Transfer, 16:255-271, 2003 Copyright © Taylor & Francis Inc. ISSN: 0891-6152 print/151-0480 online DOI: 10.1080/08916150390223092
Experimental and Numerical Method for Determining Temperature Distribution in Wood Cutting Tool J. Y. Sheikh-Ahmad∗ Department of Industrial and Manufacturing Engineering Wichita State University C. M. Lewandowski, J. A. Bailey Department of Mechanical and Aerospace Engineering J. S. Stewart Department of Wood and Paper Science North Carolina State University, Raleigh, North Carolina, 27695-7910 A 2-D boundary element model was used to solve the steady state heat conduction problem in a wood cutting tool. Cutting experiments were conducted in order to measure temperatures at remote locations on the tool rake face and at the tool/holder interface. The tool/holder interface temperatures were used as inputs into the boundary element model. Heat input into the tool was adjusted until predicted temperatures on the rake face agreed with experimental results. The proportion of the machining power conducted into the tool as heat was determined as a function of the cutting geometry, specific cutting energy and cutting speed.
The temperature of a cutting tool is one of the most important factors governing tool wear in wood machining because critical tool material properties such as hardness, toughness and chemical stability degrade with increasing tool temperatures. The contribution of various wear mechanisms to tool degradation cannot be assessed without first having accurate information on tool temperatures and how they affect the basic material properties of the cutting tool. The region near the tool cutting edge appears to be most critical. The temperature distribution near the tool edge in wood cutting has been investigated experimentally [1-14], analytically and numerically [15-19]. The tool cutting edge temperature is found to be dependent on several factors including cutting speed, feed speed, continuity of the cutting process, depth of cut and tool and workpiece materials. Cutting speed is by far the most important factor influencing cutting tool temperature. It has been shown that cutting tool temperature varies with cutting speed according to a power law [3,8,11]. Several investigators have attempted to determine the cutting tool temperature numerically [15-19]. In these works, the accuracy of the solution depended greatly on the underlying assumptions concerning boundary conditions and the proportion of the cutting energy that is converted into heat at the tool/workpiece interface. In [15] the authors adopted the well known analytical model of Lowen and Shaw [16], first developed for metal machining to calculate the proportion of cutting energy that flows into the tool as heat. In [17] the proportion of heat that flows into the tool was chosen arbitrarily to be 10%. In [18] it was assumed that the heat fluxes going into the tool and workpiece were proportional to their respective thermal conductivity and the inverse of thermal diffusivity. In [19] it was assumed that the gradient of the heat flux going into the tool and workpiece is proportional to their respective thermal conductivity. The assumptions made in these works were not verified against experimental results.
∗
Corresponding author. Department of Industrial and Manufacturing Engineering, Wichita State University, Wichita Kansas, 67260-0035, Tel: 316-978-5910, Fax: 316-978-3742, e-mail:
[email protected]. Support for this work was provided in part by North Carolina State University’s Wood Machining and Tooling Research Program through a grant from the U. S. Department of Agriculture.
In the present study, we adopt an experimental-numerical approach in order to determine the temperature distribution in the wood cutting tool. The aim of this approach is to provide a realistic measure of the proportion of cutting power that is conducted into the tool as heat. Because this quantity is not known a priori, an inverse technique is adopted where the heat flowing into the tool is adjusted in a boundary element model until temperatures on remote locations on the boundary are matched with experimental measurements. A relationship between the proportion of machining power that is conducted into the tool and the cutting speed was determined.
NOMENCLATURE ξ
dimensionless local boundary element
Γ
boundary of the domain
α
rake angle of the cutting tool, deg.
φ
temperature in the cutting tool
φo
temperature boundary condition on Γ
γ
inclination angle of the wear land with respect to the cutting velocity vector, deg.
λ
fundamental solution to the heat conduction problem in equation (1)
ρ A C c d Fc Ff Fn Ft
workpiece density wear land area constant in equation (17) workpiece specific heat depth of cut, also undeformed chip thickness cutting force frictional force on the wear land normal force on the wear land thrust force
J(ξ) k Kn(p,Q) M
Jacobian transformation thermal conductivity kernel function number of boundary elements
coordinate
over
the
n n1 n2
outward normal to the domain boundary exponent in equation (8) exponent in equation (15)
Nc(ξ) p
shape functions source point inside the boundary element domain source point on the boundary of the domain frictional power machining power specific cutting energy field point on the boundary of the domain
P Pf Pm ps Q
r(p,Q) V Vf w
heat flux boundary condition on Γ amount of power conducted into the tool as heat heat flux normal to the wear land amount of power carried away by the workpiece and the chip the distance between points p and Q cutting speed sliding speed parallel to the wear land width of the workpiece
∆φ
temperature rise of the workpiece surface
qo Qt qt Qw+c
BOUNDARY ELEMENT MODEL FORMULATION The boundary element model used in this study is based on a two dimensional elastostatic model developed by Becker [20]. Details of the model formulation are given in [20,21] and only the general outline is given here. One reason for choosing the BEM to solve this problem was its ability to give accurate results over large domains with complex shapes and large temperature gradients. In previous work [15-19] only the tip of a cutting tool was modeled numerically. It is assumed that heat conduction in the cutting tool is two dimensional, steady state, with no internal heat generation. The governing differential equation for this problem is:
∇2φ = 0
(1)
where φ is a potential function (temperature in the cutting tool) that is continuous and has first and second derivatives every where in the domain of the problem. Applying the divergence theorem [20], a boundary integral of the governing equation is obtained and can be written as:
∫
∫
φ( p ) = − K 1 ( p, Q )φ(Q)dΓ(Q ) + K 2 ( p, Q ) Γ
Γ
∂φ(Q) dΓ(Q ) ∂n
(2)
where p is a source point in the domain and Q is a field point on the boundary. The functions K1 and K2 are called the kernels and are defined by: K 1 ( p, Q ) =
∂λ( p, Q ) 1 1 and K 2 ( p, Q) = λ( p, Q) = ln ∂n 2π r ( p, Q )
(3)
where λ is the fundamental solution of the governing equation, n is the outward normal and r is the distance between p and Q. Moving the interior point p to the boundary and calling it P, equation (2) becomes:
∫
∫
C ( P)φ( P) = − K 1 ( P, Q )φ(Q)dΓ(Q ) + K 2 ( P, Q ) Γ
Γ
∂φ(Q) dΓ(Q ) ∂n
(4)
The function C(P) is dependent on the local geometry at P. Equation (4) is discretized by dividing the boundary into a number of connected elements and integrating the kernels over the boundary. The result is: C ( P)φ( P) = −
∑∑ φ(Q) ∫ K ( P, Q) N (ξ) J (ξ)dξ +∑∑ M
+1
3
M
1
m =1 c =1
3
c
m =1 c =1
−1
∫
+1
∂φ(Q ) K 2 ( P, Q ) N c (ξ) J (ξ)dξ ∂n −1
(5)
where M is the number of elements, ξ is a dimensionless local coordinate over individual boundary elements, J(ξ) is the Jacobian transformation and Nc(ξ) are shape functions. The integral functions containing the kernels can be combined into the functions Am,c and Bm,c. Equation (5) now becomes: C ( P)φ( P) = −
∑∑ φ(Q) A M
3
m,c
m =1 c =1
( P, Q ) +
∑∑ M
3
m =1 c =1
∂φ(Q) B m , c ( P, Q ) ∂n
(6)
A set of linear algebraic equations can be formed by taking each point on the boundary in turn as the load ∂φ = qo point and performing the summations in equation (6), and applying the boundary conditions φ = φ o and k ∂n on Γ. This results in the following matrix equation: ∂φ ∂n
[ A][φ] = [ B]
,
(7)
that can be solved for the unknown φ and ∂φ / ∂n using standard techniques, such as Gaussian elimination. The boundary element computer program that was developed based on this outline was verified against other solutions of standard 2D conduction problems. Because accuracy of the boundary element model predictions depended on the number of boundary elements used, a sufficiently large number of elements was utilized and the BEM solution was shown to be grid independent. PROBLEM DOMAIN AND BOUNDARY CONDITIONS
The geometry of the cutting tool representing the domain for the BEM model is shown in Figure 1. The boundary conditions for the model are applied to three regimes, namely: 1) tool wear land, 2) tool/holder interface, and 3) the remaining tool surface. A convective boundary condition with air as the convective medium was used for regime 3. The boundary condition at the tool/holder interface was determined using embedded thermocouples as described in the next section. The remaining boundary condition to be addressed is that at the tool nose region. In the present investigation, models were run to determine if there was any difference in how the temperature gradient (or heat flux) is distributed along the wear land (constant, linear, parabolic, etc.). It was found that there was very little difference in the results produced [21]. Therefore, a uniform distribution of the heat flux on the wear land was adopted.
EXPERIMENTAL WORK
Experiments were conducted to measure the temperatures at remote locations on the rake face and at the toolholder interface of a wood cutting tool for various cutting speeds. The cutting tool is a ground tungsten carbide insert 30 mm long, 12.0 mm wide and 1.5 mm thick. The cutting edge included angle was 55o. A schematic of the cutting tool and its position on the tool holder is shown in Figure 1. Type K gage 36 (0.125 mm diameter) thermocouples were used in the experimental work to measure cutting temperatures. Three thermocouples (TC1,
Mounting screws (2)
TC2 TC3
TC1 Cutting edge
Cutting insert representing BEM domain
TC4 TC5 TC6 Tool holder
Figure 1. Diagram showing geometry of the cutting tool and location of thermocouples on the center of rake face and at the tool/holdeer interface. Positions of TC1, TC2 and TC3 from cutting edge are shown in Figure 3. Positions of TC4, TC5 and TC6 from the cutting edge were 3.3, 5.8 and 9.2 mm, respectively.
TC2 and TC3) were attached to the rake face of the cutting tool using OmegabondTM 200 epoxy adhesive. The thermocouples were placed along a line perpendicular to the cutting edge at the center of the tool, and at approximately 1.0 mm intervals starting at approximately 2.0 mm from the cutting edge. The thermocouple locations were measured using a toolmakers microscope. Thermocouples were also used to measure the temperature at the tool/holder interface. Three 0.5-mm diameter holes were drilled in the tool holder such that the thermocouples could be inserted from the back side of the tool holder and brought in contact with the bottom surface of the cutting tool as shown in Figure 1. The holes were filled with thermal conductive paste and a thermocouple was placed through each hole and held securely in place. Cutting experiments were conducted on a high speed lathe as described in [21] and shown schematically in Figure 2. The workpiece material was in the form of a particleboard disk that was attached to the chuck of the lathe by a special mandrel. Cutting was carried out in such a way that the tool cutting edge was parallel to the axis of rotation of the workpiece and the feed direction was perpendicular to the axis of rotation of the workpiece. As a result, the disk diameter was continuously reduced and orthogonal cutting conditions were maintained throughout the experiment. The feed speed in all tests was maintained at 0.051 mm/rev. Table 1 provides a list of cutting conditions. Table 1. Parameter Workpiece Material
Spindle Speed Avg. Cutting Speed* Tool Geometry Feed Speed
Lathe Experiment Cutting Conditions
Value Three-layer, melamine coated particleboard disk, 750 kg/m3, 19.0 mm thickness, 381.0 mm diameter 618, 1247, 2438 rpm 8.54, 17.98, 27.56 m/s Rake: 15o. Clearance: 20o
0.051 mm/rev * Surface speed taken at the end of cutting test
Particleboard Disk
Direction of Rotation
Fc Cutting Tool
Ft
Tool Holder Spindle Shaft
Feed Direction
Dynamometer Fc Ft
Figure 2. Schematic diagram of the cutting experiment on a lathe.
A total of 14 experiments were conducted using 14 identical tools as shown in Table 2. During cutting at constant spindle speed, the cutting speed varied continuously as the disk diameter was reduced. Cutting speeds recorded in Tables 1 and 2 are those at the end of the test after cutting for a specified time. The cutting distance was calculated as the length of the spiral path taken by the tool as it traversed the rotating disk in the radial direction. Tools A1-A7 were used to measure temperatures at TC1, TC2 and TC3 and at the 3.3 mm position along the back of the tool. Tools A10-A16 were used for temperature measurement on the backside only. Thermocouple outputs were fed into signal conditioners that allowed for cold junction compensation. The cutting experiment was stopped when steady state temperatures were recorded. Along with the temperatures measurements, the cutting and thrust forces were continuously measured using a 3-component cutting force dynamometer (Kistler 9257A). A data acquisition board housed in a desktop PC was used to record the thermocouple and force signals.
Table 2. Cutting speeds and tools used in the experiments Average cutting speed* (m/s)
Tools for rake face temp measurement 8.54 A1,A2,A3 17.98 A4,A5 27.56 A6,A7 * Surface speed taken at the end of cutting test
Tools for backside temp measurement A10,A11,A12 A13,A14 A15,A6
RESULTS AND DISCUSSION 1. Experimental Results
Steady state thermocouple temperature measurements for tools A1-A7 are given in Table 3. Figure 3 shows the variation of these temperatures along the rake face of the tool for the three cutting speeds investigated. This Figure shows that temperature on the rake face increases with a decrease in distance from the cutting edge. The variation of temperature along the rake face can be described by a semi-log relationship with an average slope of 0.0298. Good repeatability of temperature measurement is generally observed for all cutting speeds, with the exception of tool A7 that seemed to register lower than normal temperatures at cutting speed of 27.56 m/s. This may be attributed to poorly bonded thermocouples and/or lower cutting forces as will be discussed later. Figure 3 also shows that rake face temperature increases with an increase in cutting speed. At any given distance along the rake face, the temperature dependence on cutting speed can be described by a power-law relationship: φ = aV n1
(8)
with an average power exponent n1 = 0.208, and the constant a depending on distance from cutting edge, as shown in Figure 4. In this Figure, temperatures given at locations where thermocouples were not physically present are obtained by linear interpolation or extrapolation of experimental data. The above mentioned findings are in agreement with the findings reported in the literature [3,8,13].
Table 3. Experimental results for tools A1-A7 Tool
TC1 (°C)
TC2 (°C)
TC3 (°C)
A1 A2 A3 A4 A5 A6 A7
157.7 155.3 154.5 196.0 184.7 201.9 197.8
147.2 148.4 146.1 184.6 175.5 189.1 183.8
139.0 140.1 139.3 172.3 164.4 173.8 167.5
Backside temp @ 3.3 mm (°C) 125.5 119.5 120.0 141.0 144.0 150.7 157.5
Fc (N)
Ft (N)
102.8 101.7 97.1 112.6 125.6 130.8 117.8
50.8 53.5 47.1 83.6 86.5 108.6 82.1
Steady state temperatures at the back of the tool (at tool/holder interface) showed similar trends to those on the rake face. The variation of these temperatures with cutting speed and distance from the cutting edge is shown in Figure 5. This Figure shows data from both sets of experiments discussed earlier and verifies the good repeatability of the testing method. The cutting force, Fc (parallel to the cutting velocity vector) and thrust force, Ft (normal to the cutting velocity vector) measured for tools A1-A7 are shown in Table 3. Both force components were obtained by averaging the force signal over a half of a second of sampling time at the end of the cutting experiments. Because forces were not collected at a common cutting distance but when a steady state temperature was reached, it is not possible to draw conclusions about the effect of cutting speed on tool forces from the available data. It is noted, however, that forces for tool A7 are lower than those for tool A6 that was used to cut a similar distance and tool A5 that was used to cut a smaller distance at a lower speed. It is believed that the lower forces for tool A7 are the reason for the lower cutting temperatures observed earlier for this tool. Table 3 also shows thermocouple temperature measurements at rake and backside of tool. The temperature measurements were used as boundary conditions in the BEM model and a solution of the steady state heat conduction problem in the tool was obtained and will be discussed in the following section. The force measurements were used to calculate energy expenditure in machining including friction, and to determine the fraction of this energy that was conducted to the tool. Because of the lower than normal forces and temperatures measured for tool A7, it was not considered in the analytical work carried out next.
500
200
400
27.56 m
/s
17.98 m
/s
8.54 m
Temperature (oC)
Temperature (oC)
250
/s
150 Tool A1 A2 A3 A4
A5 A6 A7
300 200
x = 2.2
5 mm
x = 4.5
mm
Tool
100
A1 A2 A3 A4
75
100
A5 A6 A7
50 1
2
3
4
5
Distance Along Rake Face (mm)
Figure 3. Variation of steady state temperatures along the rake face with distance from the cutting edge at various cutting speeds.
5
6
7 8 9 10
20
30
40
50
Cutting Speed (m/s)
Figure 4. Variation of steady state temperature with cutting speed at various locations on the rake face.
2. BEM Results
The BEM model developed earlier was used to solve the steady state heat conduction problem in the cutting tool. An iterative process was used in order to make BEM results match the experimentally measured temperatures on the rake face of the tool. This was accomplished by changing 1) the magnitude of the temperature gradient boundary condition on the flank wear land, and 2) the magnitude of the temperature boundary condition at position 3.3 mm from the tool nose at the back side of the tool, while leaving the temperature gradient at the tool/holder interface unchanged. The boundary conditions that resulted in agreement with the experimental data are shown in Table 4. Figure 6 shows a comparison between the temperatures on the rake face as predicted by the BEM model and the experimental results. Good agreement is generally found. The temperature distribution in the entire tool is shown in Figure 7 for a cutting speed of 27.56 m/s. Similar contour plots were obtained for the lower cutting speeds. It is noted here that the temperatures used in the model at the tool/holder interface were always slightly higher than the temperatures obtained from the experiments, as can be seen from comparison of Tables 3 and 4. A possible reason for this discrepancy could be that poor contact between the thermocouples and the backside of the tool may have resulted in temperature measurements that are lower than the actual ones. Figure 6 shows that it is possible to obtain tool nose temperatures using BEM predictions that agree with experimental results at remote locations from the tool nose. Remote temperature measurements in conjunction with BEM models are an improvement over some of the previous remote temperature measurement work. Previous work [7,8,10,13] estimated tool nose temperatures by linear extrapolation on a semi-log scale of remote temperatures measurements on the rake. Figure 6 shows that the BEM model results give much higher temperatures near the tool nose than linear extrapolation using the remote temperatures. It has been of great interest, for modeling purposes, to determine the fraction of mechanical energy that is converted into heat and conducted through the cutting tool. Since this quantity is not known a priori, the BEM analysis could not be carried out without knowledge of rake face boundary conditions. We will now consider the energy distribution between the tool, chips and workpiece based on experimental and model results. The total mechanical power expended in machining, Pm, is determined from the product of cutting force, Fc, and cutting speed, V, Pm = Fc ⋅ V
(9)
Figure 8 shows variation of machining power versus cutting speed for tool A1-A7. It can be seen that machining power increases with an increase in cutting speed according to a power relationship. A portion of this machining power is expended in overcoming friction on the wear land on the clearance face. This frictional power, Pf, is determined from the product of the frictional force, Ff, and the sliding speed, Vf, on the wear land, Pf = F f ⋅ V f
(10)
500
200
17.98 m
/s
/s
8.54 m/s
100
Tool A10 A11 A12 A1 A2 A3
A13 A14 A4 A5
A15 A16 A6 A7
3
300 27.5 6
250
17.9 8
200
A6 A7
m/ s
m/s
8.54 m/s
150
100
50 2
A1 A2 A3 A4 A5
400
Temperature (oC)
Temperature (oC)
27.56 m
4
5
6
7
8
9
10
Distance Along Back of Tool (mm)
Figure 5. Variation of steady state temperature along tool/holder interface for various cutting speeds.
0
1
2
3
4
5
6
Distance Along Rake Face (mm)
Figure 6. Comparison of BEM results and experimental results for the cutting speeds tested.
(a)
(b)
Figure 7. Temperature distribution in the entire cutting tool (a) and near the cutting edge (b) at a cutting speed of 27.56 m/s. Similar distributions were obtained for the other speeds.
For a wear land that is inclined by an angle γ with respect to the cutting velocity vector (see illustration in Figure 9), and assuming that friction on the rake face and outside the wear land on the clearance face is negligible, the frictional force and sliding speed on the wear land are determined from the expressions: F f = Fc cos γ − Ft sin γ and
(11)
V f = V sin(90 − α) / sin(90 + α − γ )
(12)
where α is the rake angle. It was determined by examining the profile of the worn tools after cutting that the wear land is inclined at approximately 10o with respect to the cutting velocity vector. The rake angle used in the experiments was 15o. Variation of Pf with cutting speed is shown in Figure 8. It can be seen from this Figure that the power expended in overcoming friction constitutes a major proportion of the machining power (82 to 87%). The difference between the machining power, Pm, and frictional power, Pf, is perhaps consumed in creating cracks and breaking fibers, which is responsible for chip formation.
Power (W)
10000 Pm Pf Qt Qw+c
1000
α Cutting edge Ff
100
Ft
γ
Fc Fn
Fr
Vc 90 − α
γ
Vf
V
10 5
10
20
30
40
50
Cutting Speed (m/s)
Figure 8. Variation of machining power, Pm, power expended in friction, Pf, heat conducted through the tool, Qt, and carried by workpiece and chips, Qw+c, with cutting speed.
Figure 9. Force and velocity diagrams at the flank wear land for an idealized worn cutting edge.
Table 4. Boundary conditions used in the BEM model Cutting Speed (m/s) 8.54 17.98 27.56
Temperature gradient at wear land (oC/mm) 600 725 875
Backside temp @ 3.3 mm (oC) 140.0 170.5 180.0
Backside temperature gradient (oC/mm) -3.653 -5.765 -6.326
The amount of energy that is transferred as heat into the cutting tool, Qt, is determined from BEM results and shown in Figure 8. The heat flux, qt, going into the tool through the wear land is calculated from the temperature gradient boundary condition values used in the BEM model and reported in Table 4. The total heat input into the tool through the wear land area, Qt, is given by:
dT ⋅A (13) dx 2 where A is the area of the wear land (approximately 0.95 mm ) and k is thermal conductivity of the tool material (100 W/m oC). Variation of Qt with cutting speed is shown in Figure 8. It is evident from this Figure that the portion of the machining power that is conducted as heat into the cutting tool is relatively small (3 to 7%), which means that the bulk of the heat generated during cutting leaves the cutting region via the chips and the workpiece surface. In a separate study [22] a temperature rise of 36o was recorded at the machined surface as it exited the cutting region. Assuming that this is also the temperature of the chip, and that the workpiece is heated to one depth of cut below the surface, the amount of heat required is calculated from the expression: Qt = qt ⋅ A = k
Qw+ c = 2∆φVwdρc (14) where ∆φ is the temperature rise of the machined surface and chip, w is the workpiece thickness, d is depth of cut, ρ is density and c is specific heat of the workpiece. This heat was found to be 55 to 72% of the machining power for a workpiece density of 776 kg/m3 and specific heat of 1357 J/kg oC, as shown in Figure 8. The choice of surface heating to one depth of cut is a first approximation based on observations by infrared radiometry of the cutting temperatures in the tool and workpiece during slow speed wood cutting [14]. The proportion of machining power that is conducted into the tool is shown in Figure 10 as the ratio Qt/Pm. This Figure also shows the ratio Qt/Pf. It is apparent from Figure 10 that Qt/Pm ≈ Qt/Pf , which means that most of the machining power is converted into heat by friction. It is also apparent that Qt/Pm decreases with an increase in cutting speed. Figure 10 indicates that previous works that relied only on frictional energy to calculate cutting tool temperatures are justified [18,19]. From the ratio Qt/Pm one can determine Qt as a function of machining power and cutting speed using the expression: Qt = Pm ⋅ CV n2 (15) where C = 0.5148 and n2 = - 0.8993. A similar equation may be obtained in terms of the frictional power. But machining power can also be determined from specific cutting energy and material removal rate as: Pm = p s ⋅ Vdw (16) where ps is the specific cutting energy for the workpiece material. Combining equations (15) and (16) one obtains: Qt = Cp s dwV n2 +1 (17) This equation can be used for predicting the proportion of energy that conducts into the tool as heat from known physical properties of the workpiece and the geometry of cutting.
CONCLUSIONS
The boundary element method was used to solve the heat conduction problem in wood cutting tool and to determine the proportion of machining power that is conducted into the tool as heat. Remote temperature measurements were used as boundary conditions in the BEM model and to verify model results. Based on experimental and numerical results it was found that:
1.
BEM is capable of solving the heat conduction problem in the cutting tool and gives more realistic predictions of the temperature distribution in the tool and the proportion of cutting power that is conducted into the tool as heat.
2.
Proportion of machining power that is conducted into the tool as heat is very small as compared to that carried away by the chips and workpiece.
3.
The proportion of machining power conducted through the tool as heat decreases with an increase in cutting speed according to a power-law relationship.
0.5
Energy Ratio
Qt/Pm Qt/Pf
0.1 0.05
0.01 5
10
20
30
40
50
Cutting Speed (m/s) Figure 10. Fraction of the machining and frictional power that is conducted into the cutting tool at various cutting speeds.
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22.
J. Sheikh-Ahmad and W. McKenzie, Measurement of tool wear and dulling in the machining of particleboard, Proc. 13th Intl Wood Machining Seminar, University of British Columbia, Vancouver, Canada, pp. 659-670, 1997.
23.
J. Holman, Experimental Methods for Engineers, McGraw Hill, 1984.
APPENDIX - UNCERTAINTY ANALYSIS
Uncertainty in the calculated powers Pm and Pf are introduced through uncertainties in the primary experimental measurements of tool forces and cutting speed. The method for estimating uncertainty presented below is based on the uncertainty analysis in [23]. For an experimental result R that is a function of a number of independent variables x1, x2, …, xn, the uncertainty in R is given by:
uR =
∂R ∂x 1
u1
2
∂R + ∂x 2
u 2
2
∂R + ..... + ∂x n
u n
2
1/ 2
(a1)
where u1, u2, …, un are the uncertainties in x1, x2, …, xn, respectively. Applying this principle to equations (9 - 13) one can obtain the individual uncertainties in the measured quantities, respectively:
u Pm = [(Vu Fc ) 2 + ( Fc u v ) 2 ]1 / 2
(a2)
u Pf = [(V f u F f ) 2 + ( F f uV ) 2 ]1 / 2
(a3)
u F f = [(cos γ ⋅ u Fc ) 2 + (− sin γ ⋅ u Ft ) 2 ]1 / 2 uV f =
sin(90 − α) uV sin(90 + α − γ )
2
V +
(a4)
sin(90 − α) cos(90 + α − γ ) u γ [sin(90 + α − γ )]2
2
1/ 2
(a5)
u Qt = [(kAu qt ) 2 + (kqt u A ) 2 ]1 / 2
(a6)
Table a1 below shows the calculated uncertainties in the various experimental results based on a 10% uncertainty in cutting and thrust forces and a 5% uncertainty in cutting speed. The uncertainties uγ, uA, and uqt were set to zero because γ, A and qt were assumed to be constants. As a result, the uncertainty in Qt was shown to be zero. Furthermore, Uncertainties in the BEM model predictions are of the same order as the uncertainty in temperature measurements, which is +/- 0.1 oC.
Table a1. Uncertainties in experimentally determined quantities Cutting Speed (m/s) 8.54 17.98 27.56
uFf
11.0 11.5 11.8
Uncertainty (%) uVf uPm
5.0 5.0 5.0
11.2 11.2 11.2
uPf
12.1 12.6 12.9
