Mathematical Formulation for the Development of Compound Curve Surface by Laser Line Heating

Transcription

1 5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India Mathematical Formulation for the Development of Compound Curve Surface by Laser Line Heating Biplab Das 1 and Pankaj Biswas 2 1 PhD Scholar, Mechanical Engineering, IIT Guwahati, India , das.biplab@iitg.ernet.in 2 Assistant Professor, Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Pin , pankaj.biswas@iitg.ernet.in Abstract Line heating assisted with laser as a heat source is a flexible forming process that forms sheet metal by means of stresses induced by external heat instead of by means of external force. The process has the potential to be applied as a primary forming method for forming accurate shapes. However successful application of this process in industry is limited due to high equipment costs and safety requirements. The production of complex shapes requires the understanding of laser-material interaction. The paper presents the mathematical formulation of development of smooth continuous curved surface. It is developed by deformation of sheet under plane stress condition by taking into account the strain distribution and the coefficient of first fundamental form of curve surface. Surface development is carried out along principal curvature direction along with the procedure for suitable determination of heating line pattern for the desired engineering surfaces. Keywords :Line heating, Strain field, Doubly curve surface, Scanning path. 1. Introduction Laser line heating is one of the most viable technique for shaping of metallic sheet components. It has a significant value to the industries that previously relied on expensive stamping processes. For past few years many approaches were taken into account in process design. Design based on Genetic algorithm(shimizu 1997; Cheng and Yao 2001) and based on response surface methodology(liu and Yao 2002) were reported.ueda et al.(1994) addresses the issue of determination of the scanning path was done on the basis of FEM by determining the in-plane strain. A suitable algorithm was developed by Jang and Moon(1998) for the determination of heating lines by calculating the line of curvature and principal curvature of the desired surface. But this method is limited to simple surfaces only. Edwardson and Watkins(1991) have developed certain rules for the positioning and sequencing of scanning paths which are required for the development of 3-D curved surface, but the work has been found to be solely dependent on the prior experience for the development of the pattern and it is found to be minimum effective when the shapes are to be formed of more complex shapes.this paper presents the mathematical approach for the development of compound curve surface by evaluating strains from the surface to its planar development corresponds to forming from planar shape to curved surface by the process of line heating. 2. Basics of differential geometry of surfaces 2.1 Regular surfaces A regular surface can be analysed in, such that: - A subset S is a regular surface if, for each ps, there exists a neighbourhood V in and a map : of an open set onto such that, x is differentiable, homeomorphism and obeys the regularity condition i.e. for each qu; the differential : is one to one. Fig. 2.1 Definition of regular surface The mapping x is called a parameterization or a system of (local) coordinates in (a neighbourhood of) p as shown in fig.2.1. The neighbourhood of p in S is called a coordinate neighbourhood. For formulating the line heating process the following algorithm has been developed, it is shown as follows:- 30-1

2 Mathematical Formulation for the Development of Compound Curve Surface by Laser Line Heating. For finding the value of we have used vector algebra. Considering a,b,c and d are vectors in, then. =..... Applying it to =. =... = (4) For regular surface, >0 everywhere, as for regular surface is not zero. From the definition of area it can be written as: Second fundamental forms For defining the curvatures of a surface S[5], considering a curve C on S passing through a point P as shown in fig. 2.3 t is the unit tangent vector, n is the unit normal vector and k is the curvature vector of the curve C at point P Fig.2.2 Flow chart for obtaining curved surface by the process of line heating The first fundamental form; Area Considering is a curve in a surface patch [3]σ such that =,, its arc length starts at a point is given by =. (1) By chain rule, = + so, = +. + = ( = ( = == + (5) The unit normal vector at each point is given by:- = (6) The normal curvature vector can be expressed as :- = (7) The scalars and are called the normal curvature and the geodesic curvature respectively. For an unit speed curve on a surface patch, its normal curvature is given by :- = +2 + (8) = +2+ (2) Where, =,=., = So, s = +2+. So by bringing inside the square root and write =, etc. we see that s is the integral of the square root of the expression: +2+ (3) This is called the first fundamental form of σ.the area of the part σ(r) of surface patch : corresponding to a region R U is Fig. 2.3 Normal Curvature Definition of normal curvature Where L= <, >, = <, > = <, >, = <, >. 30-2

3 5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India These are the coefficients of the second fundamental form. The normal curvature can also be expressed as:- = (9) Where λ = Gauss curvature The extreme values of can be obtained by evaluating =0 of Eq. (9), which gives after several algebraic manipulations: 2 +=0. (10) The values K and H are called Gauss (Gaussian) and mean curvature respectively. They are the functions of the coefficients of the first and second fundamental forms as follows: = = (11) (12) Alternatively, the Gaussian curvature K can be expressed as a function of E,F,G and their derivatives. 4 = (13) 3. Surface Development Basically surface development can be done in two ways. They are:- (i) Can be developed along iso parametric direction. (ii) Can be developed along principal curvature direction. Here surface developments based on strains along principal curvature directions are presented. As the principal curvature directions are independent of parameterization of surfaces and are unique except at umbilic points. 3.1Determination of strain field Formulation The surface is defined by the parametric vector equation r = r (u,v). The coefficients of the first fundamental form of the curved surface are given by:- =.,=.,=. (14) The strains due to development from curved surface to its plane development are, 0 and 0 along the maximum and minimum principal curvature directions respectively. Therefore the small change in length changes to 1+ and small length changes to1+, according to the definition of strain. So, =1+, =1+ (15) Where R (u,v) is the planar development. R (u,v) can also be considered as a parametric surface with its first fundamental form coefficients defined by :- =.,=.,=. (16) Since,. = +. + = (17) And. = +. + = +2 + (18) Using the relation (15) to (18), we obtain: = (19) Similarly, along minimum principal curvature direction, we have +2 + = (20) Assuming that after development the principal curvature directions remain orthogonal, this gives,. = +. + =0(21) Simplifying the above equation gives:- e =0 (22) From the three linear equations (19)-(22) in e,f,g, whose solution is given by :- e = [ ] + [ ] (23) f = [ ] [ ] (24) g = [ ] + [ ] (25) 30-3

4 Mathematical Formulation for the Development of Compound Curve Surface by Laser Line Heating We minimize the strains, and, satisfying the condition that after adding these strains to a doubly curve surface along principal curvature directions, the surface maps to a planar shape on which a Gaussian curvature is zero. This result into a minimization problem.the constrained minimization problem is discretized by using finite difference method and trapezoidal rule of integration Strain gradients After solving the strain distribution at the mid-surface there is a need for determination of the ideal gradient of the strains along normal of the mid-surface. The strain gradient provides the mechanism of surface curvature in metal forming process. Based on the Eq. (19), along s direction, the relation of the first fundamental form coefficients of the offset surface of distance d along the normal from the midsurface is: = (26) Since after development, the 2D shape is the same across the thickness, we have:- =0 (27) After substituting equation (26) into equation (27) and using expressions from the theorems on principal curvature, we have 1+. =0 (28) Expanding the above equation, we obtain at d = 0, as [ ] at d = 0 is equal to. at d = 0. which is = (29) The last equality comes from the corollary from the theorem of principal curvature. Similarly along t direction, [ ] at d = 0 is equal to. at d = 0. which is (30) 3.2 Determination of the planar developed shape Solving the non-linear minimization problem, we obtain the strains and at all grid points. The first fundamental form coefficient e,f,g of the planar developed shape is then obtained from equation (23) (25). For determination of planar coordinates, of the grid points at the corresponding planar development. These coordinates, should satisfy the following equations at all grid points:. =,. =,. = (31) After discretizing the equation (31) using finite difference method (central difference for internal points and forward and backward difference for boundary points), we obtain a system of over determined non-linear polynomial equations. Instead of solving the distance directly, we solve the following least square error unconstrained minimization problem: (32) The optimization problem can be solved by using quasi-newton method. 3.3 Heating line generation Heating paths can be determined from information of direction of principal compressive strains[6]. Heating along a line leads to bending along a direction perpendicular to that of the heating path. There is a combination of both bending and shrinkage in the heated region. When the heating paths are to be traced, the option is to be made between principal directions from either bending or shrinkage separately. The lines for bending are to be determined first, which is to be followed by shrinkage lines. Heating patterns can be obtained based on the direction of principal bending of the target surface by the use of differential geometry Bending paths The target surface is given as a function, the principal bending vectors,,, can be obtained from the Eigen value problem, that is :- =0 (33) Where k is the eigen value (principal curvature) and,,,,,are the coefficients of first and second fundamental form respectively. Once the Eigen values were obtained, which are basically the principal curvature, the corresponding Eigen vectors (principal curvature directions) can be evaluated from- = 0 (34) 0 Where is the largest principal curvature and is the corresponding eigen vector. The matrix is a singular 30-4

5 5 th International & 26 th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12 th 14 th, 2014, IIT Guwahati, Assam, India matrix as the coefficient is equal to zero. It can be written as :- + =0 (35) Or =,1, SR. (36) If for instance, the surface,, is given as a function of x and y as = + +,, Then the coefficients of the first and second fundamental form are simply:- =1+, (37) =,, (38) =1+, (39), = (40),,, = (41),,, = (42),, Where, x denotes differentiation with respect to x and, y denotes differentiation with respect to y. When the vector field for the principal directions of curvature has been calculated, it must be traced into heating lines. Heating produces more bending perpendicular to heating directions than along it. Therefore bending paths can be traced perpendicular to the field Shrinkage path The largest shrinkage direction is not necessarily perpendicular to the heating path. Differential geometry can be used, provided that a function for the target surface is known. From the strain field of the elastic analysis the principal direction, and the principal strains, can be computed from the x and y components of the strain field,, and. The z components of the strain are not needed as this is assumed to be zero in the principal direction is found from the plane strain formulation. = tan (43), = + ± + (44) Equation (43) is may be the direction of either of the two perpendicular principal directions in the plane. As the direction for is needed, and can be compared with (45) for finding the principal strains based on. The strains belong to the direction: = + + cos2 + sin2 (45) If the found is equal to the angle used in (45) was in fact the wanted. Else it belongs to the other principal strain. Previous to the tracing of strain directions into actual heating paths, requirement of distance between the heating lines must be determined. This can be explained in the following way: The displacement of heating line = (46) Where,,, are shrinkage strain, width of the heating line, elastic analysis strain and distance between line of interest. Thus heating line can be traced. 4 Conclusion The fabrication of the curved surface starts from the development of unfolded flat configuration. Here line heating is used as a forming method; here stretching and shrinkage of the surface are to be taken into account as because of the development of nondevelopable surface. Here in this paper step by step problem solving procedure has been discussed for development of doubly curved surface mathematically, with an approach of using differential geometry as a tool. The contribution of this paper includes:- Development and design of doubly curve surface. Step by step problem solving procedure, mathematically, with the involvement of mechanical parameters involved in the process of forming (laser line heating). 5 References Patrikalakis, N.M. and Maekawa, T.(2002), Shape interrogation for Computer Aided Design and Manufacturing, First edition, Springer-Verlag, Heidelberg, Germany. Rogers,D.F. and Adams, J.A.(2003), Mathematical Elements for Computer Graphics, Second Edition, Tata McGraw-Hill Publication, New Delhi. 30-5

6 Mathematical Formulation for the Development of Compound Curve Surface by Laser Line Heating Pressley, Andrew (2001), Elementary Differential Geometry,Spinger Undergraduate Mathematics Series ISSN ,Springer-Verlag London Limited. docarmo, Manfredo P. (1976),Differential Geometry of Curves and Surfaces,Prentice-Hall, Inc., Englewood Cliffs, New Jersey. curved surfaces, Computer Aided Geometric Design Vol. 17 pp Clausen, H.B.(2000), Plate forming by line heating, PhD thesis, Department of Naval Architecture and Offshore Engineering, Technical University of Denmark. Yu, Guoxin, Patrikalakis, Nicholas M., Maekawa Takashi (2000), Optimal development of doubly. 30-6