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- Lorong P., Larue A., Duarte A.P. Dynamic Study of Thin Wall Part Turning.Advanced Materials Research, 2011, vol. 223, pp. 591{599. DOI:10.4028/www.scientific.net/AMR.223.591

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- Nevertheless, in some cases, the use of dynamometers can be problematic, for instance in case of thin-walled workpieces in presence of instabilities: due to the presence of resonances in the frequency response of the dynamometer itself can induce significant perturbations in the measured signals. This was the case for
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- : the addition of the dynamic system of the dynamometer can modify the conditions of the chatter onset. In the present paper, we address the problem of quasi-static evaluation of the cutting force during turning cylindrical shells.

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- Fang N., Wu Q. A Comparative Study of the Cutting Forces in High Speed Machining of Ti-6Al-4V and Inconel 718 with a Round Cutting Edge Tool.Journal of Materials Processing Technology, 2009, vol. 209, no. 9, pp. 4385{4389. DOI:10.1016/j.jmatprotec.2008.10.013

Total in-text references: 2- In-text reference with the coordinate start=5408
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- The shell is subjected to pin-load, which is represented by the two components of the cutting force: radialFRand curcumferentialFC, as shown in Fig.2. We neglect the axial component of the cutting force because it is always much smaller than the other two components
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- , and because the stiffness in thez-direction is much higher than in the other two directions. ForcesFRandFC act on the point with coordinates(sF,0). We introduceΛas a changeable dimensionless parameter so thatsF= ΛL, see (fig.2).

- In-text reference with the coordinate start=13094
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- In our work, we have set the relative accuracy toε= 0.01 = 1%. Now that we know how to calculate matrixA, we can estimate the order of magnitude of the displacements. The magnitude of the cutting force is approximately102{103N, according to
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- . Using this data and our model, we have calculated that magnitudes of the displacements are within10−5m. 3. Optimization of the displacement sensors location We need to determine the best set of parametersΦ1andΦ2(the displacement sensors angular location) that would make the system (2) as well-conditioned as possible.

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- Biderman V.L. Mekhanika tonkostennykh konstruktsiy [Mechanics of thin-walled structures]. Moscow, Mashinostroenie, 1977. 488 p. (In Russian)

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- Given that the shell's geometry and material properties are not subject to variation, it is apparent that the components of matrixAare dependant on the three variable parameters that we have introduced earlier:A=A(Φ1,Φ2,Λ). 2. Calculation of the flexibility matrix According to
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- , the general system of equations for a Kirchhoff-Love thin-walled cylindrical shell, which is shown in Fig.1, can be written in the following form ∂u ∂s = 1−ν2 Eh T1− ν R (∂v ∂φ +w ) , ∂v ∂s = 2(1 +ν) Eh S∗1− h2 3R2 ∂θ1 ∂φ + (h2 3R3 − 1 R )∂u ∂φ , ∂w ∂s =−θ1, ∂θ1 ∂s = 12(1−ν2) Eh3 M1+ ν R2 (∂2w ∂φ2 − ∂

- In-text reference with the coordinate start=11056
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- are the boundary conditions: ( u(k), v(k), w(k), θ1(k) ) = 0ats= 0; ( RT1(k), RS∗1(k), RQ∗1(k), RM1(k) ) = 0ats=L. (7) Here are the continuity conditions at point (s=sF): y(k)(sF+ε) =y(k)(sF−ε) + 1 2π (0,0,0,0,0,−FC, FR,0), whereεis an infinitesimally small parameter. System (6) along with boundary conditions (7) represent a boundary value problem. We have used the initial parameters method
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- in order to solve this problem by means of numerical integration. The Godunov orthogonalization method [5] was incorporated to ensure numerical stability of the solution. Moreover, the method was further modified in order to eliminate the well-known Gram-Schmidt process's weakness [6].

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- Godunov S.K. [Numerical solution of boundary-value problems for systems of linear ordinary differential equations].Uspekhi matematicheskikh nauk, 1961, vol. 16, no. 3, pp. 171{174. (In Russian)

Total in-text references: 1- In-text reference with the coordinate start=11162
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- System (6) along with boundary conditions (7) represent a boundary value problem. We have used the initial parameters method [4] in order to solve this problem by means of numerical integration. The Godunov orthogonalization method
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- was incorporated to ensure numerical stability of the solution. Moreover, the method was further modified in order to eliminate the well-known Gram-Schmidt process's weakness [6]. The Gram-Schmidt process was replaced by the Householder transformation [7], which effectively performs the same thing | orthonormalizes a set of vectors in the Euclidean spaceRn.

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- Higham N.J. Accuracy and Stability of Numerical Algorithms. 2nded. Philadelphia, SIAM. 2002.

Total in-text references: 3- In-text reference with the coordinate start=11339
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- The Godunov orthogonalization method [5] was incorporated to ensure numerical stability of the solution. Moreover, the method was further modified in order to eliminate the well-known Gram-Schmidt process's weakness
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- [6]
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- . The Gram-Schmidt process was replaced by the Householder transformation [7], which effectively performs the same thing | orthonormalizes a set of vectors in the Euclidean spaceRn. Harmonicw(k)(L)that corresponds to the radial displacements of points located on the free end of the shell, can be represented as a linear combination of the cutting force components: w(k)(L) =FC·α(k)+FR·β(k), where co

- In-text reference with the coordinate start=13511
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- Optimization of the displacement sensors location We need to determine the best set of parametersΦ1andΦ2(the displacement sensors angular location) that would make the system (2) as well-conditioned as possible. A measure of a square matrix's numerical stability is called its condition numberμ. By definition
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- , μ(A) =‖A‖·‖A−1‖. Condition numberμis always a positive number, and it cannot be less than one. The closer the condition numberμof the flexibility matrixAis to one, the more well-conditioned this matrix is.

- In-text reference with the coordinate start=15324
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- ◦66.6◦1.5 DS77.7◦77.7◦2.8 ES89.3◦89.3◦1.9 F1119.4◦83.5◦2.4 F283.5◦119.4◦2.4 GS136.2◦136.2◦2.7 HS149.5◦149.5◦1.7 Table 3 Local extrema off CodeΦ1Φ2f I1208.3◦137.5◦1.8 I2222.5◦151.7◦1.8 J1241.1◦83.5◦2.0 J2276.5◦119.9◦2.0 K1268.5◦78.6◦2.0 K2281.5◦91.5◦2.0 L1293.5◦25.3◦1.4 L2334.7◦66.5◦1.4 M1329.3◦20.5◦1.2 M2339.5◦30.7◦1.2 It means that at the worst case scenario we could lose 1{2 significant digits
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- calculating the components of the cutting force using equation (2), which corresponds to relative accuracyε= 10−10. Such loss is not significant in comparison with other sources of error in our model.

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- Householder A.S. Unitary Triangularization of a Nonsymmetric Matrix.Journal of the ACM, 1958, vol. 5, no. 4, pp. 339{342. DOI:10.1145/320941.320947

Total in-text references: 1- In-text reference with the coordinate start=11415
- Prefix
- The Godunov orthogonalization method [5] was incorporated to ensure numerical stability of the solution. Moreover, the method was further modified in order to eliminate the well-known Gram-Schmidt process's weakness [6]. The Gram-Schmidt process was replaced by the Householder transformation
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- [7]
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- , which effectively performs the same thing | orthonormalizes a set of vectors in the Euclidean spaceRn. Harmonicw(k)(L)that corresponds to the radial displacements of points located on the free end of the shell, can be represented as a linear combination of the cutting force components: w(k)(L) =FC·α(k)+FR·β(k), where coefficientsα(k)andβ(k)depend only on parameterΛand have been obtained after nu

- In-text reference with the coordinate start=11415