An Exploratory Study of Taylor’s Tool-Life Equation by Power Transformations

1966 ◽  
Vol 88 (1) ◽  
pp. 81-89 ◽  
Author(s):  
S. M. Wu ◽  
D. S. Ermer ◽  
W. J. Hill
Keyword(s):  
Tool Life ◽  
Exploratory Study ◽  
General Class ◽  
Logarithmic Transformation ◽  
Carbide Tool ◽  
Independent Variables ◽  
Life Data ◽  
Power Transformations ◽  
Best Fit ◽  
Special Case

Transformations of both dependent and independent variables are employed to investigate the linearization of Taylor’s tool-life equation. This exploratory study indicates that a logarithmic transformation, which is a special case of the general class of power transformations, gives the best fit for HSS tool-life data. However, the study does not show that a logarithmic transformation is the best for carbide tool-life data. For a wide cutting range, where Taylor’s tool-life equation does not hold, a linear equation instead of a second-order relationship for the prediction of tool life can be determined by the proper transformation.

scholarly journals The unified Riemann-Liouville fractional derivative formulae

2005 ◽  
Vol 36 (3) ◽  
pp. 231-236
Author(s):  
R. C. Soni ◽  
Deepika Singh
Keyword(s):  
Fractional Derivative ◽  
General Class ◽  
Bessel Functions ◽  
Jacobi Polynomials ◽  
Independent Variables ◽  
General Class Of Polynomials ◽  
Derivative Formula ◽  
Liouville Fractional Derivative ◽  
Biorthogonal Polynomials ◽  
Special Case

In this paper, we obtain two unified fractional derivative formulae. The first involves the product of two general class of polynomials and the multivariable $H$-function. The second fractional derivative formula also involves the product of two general class of polynomials and the multivariable $H$-function and has been obtained by the application of the first fractional derivative formula twice and it has two independent variables instead of one. The polynomials and the functions involved in both the fractional derivative formulae as well as their arguments are quite general in nature and so our findings provide interesting unifications and extensions of a number of (known and new) results. For the sake of illustration, we point out that the fractional derivative formulae recently obtained by Srivastava, Chandel and Vishwakarma [11], Srivastava and Goyal [12], Gupta, Agrawal and Soni [4], Gupta and Agrawal [3] follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Konhauser biorthogonal polynomials, the Jacobi polynomials and the product of $r$ different modified Bessel functions of the second kind as a simple special case of our first formula.

Investigation on the Tool Wear Model and Equivalent Tool Life in End Milling Titanium Alloy Ti6Al4V

2018 ◽  
Author(s):  
Kai Guo ◽  
Bin Yang ◽  
Jie Sun ◽  
Vinothkumar Sivalingam
Keyword(s):  
Tool Wear ◽  
Titanium Alloys ◽  
Tool Life ◽  
End Milling ◽  
Cutting Speed ◽  
Carbide Tool ◽  
Wear Model ◽  
Wear Process ◽  
Coated Carbide Tool ◽  
Coated Carbide

Titanium alloys are widely utilized in aerospace thanks to their excellent combination of high-specific strength, fracture, corrosion resistance characteristics, etc. However, titanium alloys are difficult-to-machine materials. Tool wear is thus of great importance to understand and quantitatively predict tool life. In this study, the wear of coated carbide tool in milling Ti-6Al-4V alloy was assessed by characterization of the worn tool cutting edge. Furthermore, a tool wear model for end milling cutter is established with considering the joint effect of cutting speed and feed rate for characterizing tool wear process and predicting tool wear. Based on the proposed tool wear model equivalent tool life is put forward to evaluate cutting tool life under different cutting conditions. The modelling process of tool wear is given and discussed according to the specific conditions. Experimental work and validation are performed for coated carbide tool milling Ti-6Al-4V alloy.

scholarly journals Estimation of Regression Parameters from Noise Multiplied Data

2013 ◽  
Vol 4 (2) ◽  
Author(s):  
Yan-Xia Lin ◽  
Phillip Wise
Keyword(s):  
Regression Analysis ◽  
Regression Model ◽  
Real Life ◽  
Simulation Studies ◽  
Independent Variables ◽  
Multiplicative Noises ◽  
Life Data ◽  
Regression Parameters ◽  
Data Application ◽  
Real Life Data

This paper considers the scenario that all data entries in a confidentialised unit record file were masked by multiplicative noises, regardless of whether unit records are sensitive or not and regardless of whether the masked variables are dependent or independent variables in the underlying regression analysis. A technique is introduced in this paper to show how to estimate parameters in a regression model, which is originally fitted by unmasked data, based on masked data. Several simulation studies and a real-life data application are presented.

On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales

1968 ◽  
Vol 34 (1) ◽  
pp. 145-158 ◽  
Author(s):  
K. Kuen Tam
Keyword(s):  
Exact Solution ◽  
Velocity Profile ◽  
Asymptotic Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Order Equation ◽  
Zeroth Order ◽  
Independent Variables ◽  
Sommerfeld Equation ◽  
Special Case

The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.

Comparative assessment between AlTiN and AlTiSiN coated carbide tools towards machinability improvement of AISI D6 steel in dry hard turning

2021 ◽  
pp. 095440622110373
Author(s):  
Anshuman Das ◽  
Miyaz Kamal ◽  
Sudhansu Ranjan Das ◽  
Saroj Kumar Patel ◽  
Asutosh Panda ◽  
...  
Keyword(s):  
Cutting Force ◽  
Tool Life ◽  
Hard Turning ◽  
Flank Wear ◽  
Chip Morphology ◽  
Comparative Assessment ◽  
Carbide Tool ◽  
Crater Wear ◽  
Coated Carbide ◽  
Coated Carbide Tools

AISI D6 (hardness 65 HRC) is one of the hard-to-cut steel alloys and commonly used in mould and die making industries. In general, CBN and PCBN tools are used for machining hardened steel but its higher cost makes the use for limited applications. However, the usefulness of carbide tool with selective coatings is the best substitute having comparable tool life, and in terms of cost is approximately one-tenth of CBN tool. The present study highlights a detailed analysis on machinability investigation of hardened AISI D6 alloy die steel using newly developed SPPP-AlTiSiN coated carbide tools in finish dry turning operation. In addition, a comparative assessment has been performed based on the effectiveness of cutting tool performance of nanocomposite coating of AlTiN deposited by hyperlox PVD technique and a coating of AlTiSiN deposited by scalable pulsed power plasma (SPPP) technique. The required number of machining trials under varied cutting conditions (speed, depth of cut, feed) were based on L16 orthogonal array design which investigated the crater wear, flank wear, surface roughness, chip morphology, and cutting force in hard turning. Out of the two cutting tools, newly-developed nanocomposite (SPPP-AlTiSiN) coated carbide tool promises an improved surface finish, minimum cutting force, longer tool life due to lower value of crater & flank wears, and considerable improvement in tool life (i.e., by 47.83%). At higher cutting speeds, the crater wear length and flank wear increases whereas the surface roughness, crater wear width and cutting force decreases. Chip morphology confirmed the formation of serrated type saw tooth chips.

scholarly journals Symmetric Determinants and the Cayley and Capelli Operator

1948 ◽  
Vol 8 (2) ◽  
pp. 76-86 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Differential Operators ◽  
Matrix Operator ◽  
Linear Combinations ◽  
Classical Invariant Theory ◽  
Independent Variables ◽  
The Matrix ◽  
Classical Invariant ◽  
Initial Discovery ◽  
Special Case

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

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