Numerical Method to Calculate the Deformation of ... - ScienceDirect

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ScienceDirect Procedia Engineering 132 (2015) 872 – 879

The Manufacturing Engineering Society International Conference, MESIC 2015

Numerical Method to Calculate the Deformation of Thin Rings during Turning Operation and its Influence on the Roundness Tolerance M. Estremsa,*, M. Arizmendia, A. J. Zabaletaa, A. Gilb a

Dept. of Mechanical Eng. TECNUN. Univ of Navarra. Paseo Manuel Lardizábal, 13. San Sebastián. Spain Public University of Navarra, Department of Mechanical, Energy and Materials Engineering, Edificio Departamental Los Pinos, Campus Arrosadía, Pamplona. Spain.

b

Abstract This paper presents a model to estimate the total deformation of turned rings. It takes into account the clamping forces values at different angular positions of the workpiece, and the cutting forces during machining. The model is based on the Castigliano Theorem and uses the Chebyshev polynomials to get a stable solution to the differential equation of deformation. Cutting tests were conducted in a lathe and resultant roundness profiles were measured. The results showed good agreement with the model predictions. © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license © 2015 2016The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of MESIC 2015. Peer-review under responsibility of the Scientific Committee of MESIC 2015 Keywords: Clamping Forces, Turned Rings, Roundness Error, Chebyshev polynomials, Castigliano Theorem

1. Introduction Thin rings are common parts in gearboxes, motors, gear rims, pulleys, etc., which are spread through the automotive and aeronautical sector. These rings are required with a high degree of roundness and tight tolerances. A major manufacturing difficulty is to minimize the effect of clamping forces on the highly flexible workpiece. Moreover, many rings in the "as received" condition present already important roundness errors when placed on a

* Corresponding author: Tel.: +34-943-219-877; fax: +34-943-311-422. E-mail address: [email protected]

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of MESIC 2015

doi:10.1016/j.proeng.2015.12.572

M. Estrems et al. / Procedia Engineering 132 (2015) 872 – 879

lathe chuck for a finishing operation. These errors come usually from a thermal treatment, and/or plastic deformation in previous production phases, etc. But the main source of error comes from the deformation during the fixturing of the ring during machining. The clamping forces of the jaws over a highly flexible ring causes a ring deformation that creates roundness errors in the workpiece. Therefore, the target is to diminish the clamping force as much as possible but, at the same time, avoiding workpiece sliding on the chuck jaws. In the literature, some formulations and methods are provided for calculations of deformation in rings placed on jaw chucks. The most common method is the use of the Finite Element Method (FEM) [1], but it has a high computational cost because it requires a fine mesh to achieve sufficient precision. One analytical method is that developed by Malluck and Melkote [2] based on some formulae of an old book from which the expressions of the internal moment and the ring deformation are deduced. This method was checked experimentally and with the Finite Element Method (FEM). Since the analytical method is quicker than that of the (FEM), many papers are based in it. For example, Kurnady et al. [3] used it in optimization of the machining operation. Sölter et al. [4, 5], used it in a model to predict errors using different fixturing strategies. Stöbener et al. [6] used it as an analytical tool to study different strategies such as ultrasonic sensing and the use of Fast Tool Servo for the compensation of errors in real time. Soleiman and Mehr [7] uses this approach also, etc. But one shortcoming of this model is that the derivation of those formulae employed by Malluck and Melkote [2] is not accessible to the scientific community. Another shortcoming is that they assume a) a perfect roundness of the “as received” ring preform, b) constant thickness of the ring, c) perfect homogeneity of the work material, d) the absence of residual stress, etc. These non-homogeneous characteristics can be added to the problem by a discretization of the problem in one dimension, i.e. putting the points (with their own characteristics) in a row in which the first point and the last one are the same, as corresponds to a close ring. In this way, the problem dimension would not be as big as the one of a FEM, and will be easy to implement, quicker to calculate, and simpler to check and avoid errors. In this work, a simpler and even more accessible method than that of Malluck and Melkote [2] is proposed, that takes into account the previous deformation of the ring, and the small variation of thickness. An original method has been developed to solve numerically the differential equation of deformation that allows coherent boundary conditions with the close form of the geometry. The whole problem is stable and easy to integrate in applications of optimization of integration with sensors. The method has been checked experimentally. 2. Model of ring deformation The model presented uses the Castigliano’s Theorem for calculating the bending moments along the ring perimeter, and it treats the problem as two-dimensional with the hypothesis of the low influence of radial and axial force on the flexural moments. The «as received» ring geometry is always distorted so, in this paper the final deformation is calculated by using the actual deformed ring curvature radii instead of considering that it has a constant radius, as it has been done so far in literature, which increases the non-homogeneity and asymmetry of bending moments. Then the differential equation of curved bars derived by Timoshenko [9] is used to calculate the deformation of the ring. This equation is solved by the previous transformation in Chebyshev polynomials [10]. In this way, the differential equation is transformed into an algebraic one. It allows setting the boundary condition values by making equals both the deformations at initial point and at the final one. This is another aspect not addressed so far in the literature. 2.1. Castigliano’s Theorem to calculate the moments in the ring From the sector of the ring, according to Figure 1, the balance equations are:

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MI QI NI

M 0  Q0 r sin I  N 0 r 1  cosI  6i Pi cos2Ii  cosI  Ii

N 0 sin I  Q0 cosI  6i Pi cosI  Ii N 0 cosI  Q0 sin I  6i Pi sin I  Ii

(1)

The clamping forces Pi are introduced only if acting on the sector of the ring studied. The equilibrium conditions, the symmetry of the problem, the uniformity of the ring and de absence of friction, result in that all three Pi’s are equal in magnitude. But, when cutting forces (Fc, Ft) are applied, friction forces are generated in the contact points of the jaws with the workpiece Ti’s, see Figure 2, and the normal forces Pi are no longer equal. The equilibrium of torques and forces gives a hyperstatic set of equations and therefore in order to calculate the resultant Ti’s and Pi’s we need an added condition.

Fig. 1. Forces acting on the ring, and a sector of this one.

The conditions are given by the deflections values in the ring-jaws contact points, in a similar way as was proposed by Walter et al. [11]. As the flexibility of the ring is much higher than that of the jaws, the contact rigidity calculation is based on the deformation of the ring at an angle of 2/3π, as shown in Figure 3. We assume all jaws have identical contact normal rigidity kn and tangential rigidity kt . The rigidity of each jaw has been calculated using the Theorem of Castigliano as in [12], through the flexibility of the jaw point when a magnitude of force unity is applied in both axial and radial directions (Figure 3). In this way, Pi and Ti could be expressed as a function of the δx, δy and δθ deformations of the rigid solid as it is now considered to be the ring.

Pi

P0  kn (G x sin Ii  G y cos Ii )

Ti

kt (G x cos Ii  G y sin Ii  rGT )

Fig. 2. Forces acting on the ring, and a sector of this one.

(2)

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Fig. 3. Jaws rigidity calculation

Applying the equilibrium of forces in the ring, the following equations are obtained:

GT



Fc ;G x 3kt

Fc ;G y 1.5(kn  kt )

Fc 1.5(kn  kt )

(3)

By taking (3) and (2), Ti and Pi can be calculated. Although these expressions has been obtained for a position of the cutting force Fc at 90°, deformation for other positions of Fc can be calculated decomposing this force in both directions. By including the cutting Fc and the tangential forces Ti at the jaws, equations (1) can be re-written adding several terms as follows:

MI

} Ti r (1  cos(I  Ii ))

QI

} Ti sin(I  Ii )

NI

} Ti cos(I  Ii )

(4)

The cutting force would be considered as another Ti with its corresponding angle ϕc. M0, Q0 and N0, would be calculated through the three conditions given by the continuity in point 0 in Figures 2 and 3. They are: that for ϕ=2π, then δx=0, δy=0 and δθ=0. To calculate δx, δy and δθ, the Castigliano’s Theorem is used. First, as if M0=0, Q0=0 and N0=0, and then obtaining an analytical relation between the constant of integration and the M0, Q0 and N0 values. As deformations, both previous and at machining are small, the radius and the ring radial area can be considered as constant. From the value of the strain energy, Uf, of the ring in sector [0, ϕ] of Figure 1:

2U f

³

I1

0

§ M I2 kQI2 NI2 ·   ¨¨ ¸¸rdI EI GA EA z © ¹

(5)

Where E is the Young modulus, G is the Shear modulus, Iz is the moment of inertia of the section, and A is the area of the ring. Mϕ, Qϕ and Nϕ can be obtained from the equations (1) and (4). According to the Castigliano’s Theorem the displacements δx and δy, and the torsion δθ are the derivatives of the function Uf with the forces and torques acting in such a point. In point 0, when ϕ1=2π, then

G y0 G y0

wU f

wQ0 0; G x 0

;G x 0

wU f

wN 0

0;GT 0

;GT 0

wU f

wM 0

(6)

0

Solving the three integrals gives constants of integration C1, C2 and C3 that can be obtained through the three conditions, giving a set of equations from which we get the values of M0, N0 and H0, and then, the values of Mϕ, Qϕ and Vϕ in equations (1) and (4).

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ª C1 º «C » « 2» «¬ C3 ¼»

ª 2S r « « EI z « « 0 « « « 2S « EI ¬ z

§ 3r 2 k 1 ·º S ¨   ¸» © EI z GA EA ¹ » » ªM 0 º § 3r 2 k 1 · » « N0 » S ¨   0 ¸ EI GA EA »« » © z ¹ » ¬« H 0 ¼» 2S r 2 » 0 » EI z ¼ 0

(7)

Applying equation (1) the bending moment Mϕ at any ring section can be calculated and used as input in the differential equation of the deformation of the ring y [9].

ycc  y

 M I r (I )2

(8)

EI z

2.2. Chebyshev polynomials to solve differential equation To solve the differential equation (8) the method described in [10] has been employed. It computes the coefficients of the approximation of Chebyshev of the function y by solving an algebraic equation. This method has been chosen because it allows setting boundary conditions in closed loop systems, it provides more stable and robust solutions, and it is less sensible to different discretization parameters. Functions approximated by Chebyshev polynomials have the advantage of having a matrix form and its derivatives can be obtained in a similar way. y( x) T( x) A (9) Where T(x) is the Chebyshev polynomial in the interval [-1, 1], and A is the vector of coefficients of the approximation as shown in equation (10).

T( x) [T0 ( x),T1 ( x),},TN ( x)] A

(10)

[a0 / 2, a1 ,}, aN ]T

The coefficients of the derivative function can be obtained from the function through a square matrix M:

yc( x)

T( x)(2MA)

ycc( x)

T( x)(4M 2 A)

(11)

The matrix M for an even number of terms is:

M

ª 0 1/ 2 0 3 / 2 0 5 / 2 «0 0 2 0 4 0 « «0 0 0 3 0 5 « «} } } } } } «} } } } } } « ¬} } } } } }

} }

}

}

}

}

} º } } } } } } } »» } } } } } } } » » } } } } } } } » } 0 N 6 0 N 4 0 N  2» » } 0 0 N 5 0 N 3 0 ¼

(12)

The independent term of the equation (8) can also be approximated by Chebyshev polynomials with a coefficient matrix F. Thus the differential equation is transformed into an algebraic equation whose unknowns are the

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coefficients of Chebyshev of the A matrix. The term π2 comes from the application of the chain rule, since the angle interval [0, 2π] is different from the Chebyshev approximation interval [-1, 1].

§ 4M 2 · ¨ 2 I¸A F © S ¹

(13)

The boundary conditions are introduced by doing: y(-1) = y(1) by replacing the last two rows of the system by Ti (-1) and Ti (1) and putting an identical value in the two last rows of the independent term of the equation (13). This value is chosen such that the average of the result y(x) in the whole circumference is zero. This value is calculated by the secant method. 3. Effect of the clamping force on the bending moments and deformations First we will study the effect of clamping force disregarding the cutting forces. The method has been applied to a ring of Aluminum of 80 mm in diameter, 4.5 mm thickness and 10 mm width, with a Young modulus E=7e10 MPa and a Poison ratio of ν=0.33. The normal force applied externally is 980 N in each jaw. The objective would be to machine the interior of the ring. In Figure 4 the bending moment and radial deformations are represented along the ring perimeter, showing singularities in the contact points between the jaws and the ring. The compressive force of the section is N0=-536 N, the maximum moment is M0=3.48 N·m, and it occurs at 0, 2π/3, 4π/3 radians, and the minimum moment is Mi=-6.62 N·m, which occurs at the jaw position. Shear force is V0=0 in all sections.

Fig. 4. Bending moment and radial deformations of the ring with a clamping force of 930 N

Fig. 5. Ring shape evolution from original to final stages.

This moment diagram is coherent with the system as is has three identical intervals. The moment at the beginning of the ring is identical to that of the end point with continuity C1. Using the Navier distribution of stress, in the jawworkpiece contact points there are compressive stresses in the internal surface and compression in the external ones, but in the middle points (where maximum moments happens) the tensional stress are in the external surface.

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Figure 4 shows also the deformation y resulting of the resolution of the differential equation (8). As it is observed, the intervals are not exactly identical due to the triple symmetry of the real problem in the interval [0, 2π], that does not coincide to the simple symmetry of the Chebyshev interpolation in the interval [-1, 1]. Moreover it is observed that the maximum deflection happens in the jaw-ring contact location where more material would be removed. When the internal surface is machined, the resultant thickness at the jaw-ring contact points would be minimal. Moreover, the machined internal surface would be more relaxed, so when the piece is released, the machined surface would deform much more than that of external surface that has not be modified. This would translate in a cylindrical form in the external surface as in the beginning, and a mirror deformed of the internal surface as it was clamped. This is illustrated in Figure 5 in an amplified view.

Fig. 6. Evolution of the tangential force in the jaws and the radial deformation at the cutting point when the clamping force is Pi=430 N.

Fig. 7. Deformation of the ring (mm) theoretical and experimental with a clamping force of 930 N.

4. Effect of cutting force Finally an analysis of the effect of cutting force has been made. If we consider the machining of an Aluminum ring, with tool of finishing operation Sandvik DCMT 11T308EN-ZM, a feed f=0.1 mm/rev depth of cut ap=0.5 mm, and a cutting speed of 60 m/min, then if the specific cutting force for Aluminum is ps= 300 N/mm2, the resultant cutting force would be Fc=300∙0.1∙0.5=15 N. To this rotating force the jaws respond with the tangential reactions represented in Figure 6. The symmetry of the problem shows that the curves could be consecutive covering the

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whole circumference. The values represented are relative to the initial clamping force, so the limits of friction force remains well limited below the minimum. The radial force Ft is depreciated because it is lower by far than the clamping forces. Another interesting result is the calculation of the radial displacement in the point where the cutting force is acting and the new surface is generated. Figure 6 shows this result and although a small instability is observed, the sinusoid form can be easily recognized. 5. Experimental validation The method has been applied to a ring of Aluminum of 80 mm in diameter, 4.5 mm thickness and 10 mm width. The ring has been externally clamped with 3 jaws and 4 different normal force values and has been machined with a depth of cut of 0.5 mm and a feed of 0.1 mm in order to reduce the cutting force value. After machining, the part has been unclamped and its deviation of the deformed shape was measured at 3 different depths. As it is seen in Figure 7, profiles generally agree well in form. The differences between theoretical and experimental deformations, shown in Figure 7, can be due to several factors: a) the estimation of the material constants, b) the difficulty of obtaining precise gripping forces values, and c) the initial inaccuracy of the ring that amplifies the deformation when three equal loads are applied. 6. Conclusions A numerical method for the calculation of deviation of the deformed shape of turned rings that takes into account their “as received” shape, the cutting force values and the thickness variation of the ring, has been developed. It is fast and can be useful in optimization problems. Moreover, it can be used in combination with other strategies as the use of Fast Tool Servos, in order to improve the accuracy of ring parts. Acknowledgements The authors would like to thank the Basque Government for supporting this work made under the ETORTEK program within the MARGUNE CRC framework. References [1] U. Heisel, C. Kang. Model-based form error compensation in the turning of thin-walled cylindrical parts. Production Engineering, Vol. 5 (2011), pp. 151-158. [2] J. Malluck, S. N. Melkote, Modelling of deformation of ring shaped workpieces due to chucking and cutting forces. Trans. ASME, Journal of manufacturing science and engineering, Vol. 126 (2004), pp. 141-147. [3] M. S. Kurnadi, J. Morehouse, S. N. Melkote, A workholding optimization model for turning of ring-shaped parts. The International Journal of Advanced Manufacturing Technology. Vol. 32 (2007), pp. 656–665. [4] J. Sölter, C. Grote, E. Brinksmeier, Prediction of Clamping-Induced Roundness Deviations of Turned Rings. Proc. 12th CIRP Conference on Modelling of Machining Operations, San Sebastián (SPAIN), Vol. 2, pp. 703-709, 2009. [5] L. Nowag, J. Sölter, E. Brinksmeier. Influence of turning parameters on distortion of bearing rings. Production Engineering, Vol. 1(2007), pp. 135-139. [6] J. Sölter, C. Grote, E. Brinksmeier. Influence of clamping strategies on roundness deviations of turned rings. Machining Science and Technology, Vol. 15(2011), pp. 338-355. [7] D. Stöbener, B. Beekhuis, Application of an in situ measuring system for the compensation of wall thickness variations during turning of thinwalled rings. CIRP Annals - Manufacturing Technology, Vol. 62 (2013), pp. 511-514. [8] H. Soleimanimehr, M.J. Nategh, H. Jamshidi. Mechanistic Model of Work-Piece Diametrical Error in Conventional and Ultrasonic Assisted Turning. Advanced Materials Research Vol. 445 (2012), pp. 911-916. [9] S. Timoshenko, Strength of Materials. Part II. (2nd Edition) Ed. Van Nostran Company Inc., 1947, pp. 102-103. [10] M. Sezera, M. Kaynak, Chebyshev polynomial solutions of linear differential equations. International Journal of Mathematical Education in Science and Technology, Vol. 27 (1996), pp. 607-618 [11] M. F. Walter, J. E. Ståhl, The connection between cutting and clamping forces in turning. International Journal of Machine Tools and Manufacture, Vol. 34 (1994), pp. 991-1003. [12] W.C. Young, R.G. Budynas, Roak’s Formulas for Stress and Strain. McGraw-Hill, 7th Edition. (2002), p. 281