A New Method for Production of Double-Curvature Dome-Shaped Part: Statistical Analysis of the Effects of Process Parameters on the Radius of Curvature of Obtained Dome Shape
Authors: Safari Mehdi , Farzin Mahmoud 
Published: April 2018

In this work, a new method is proposed for production of double-curvature domeshaped part. Also, the effects of process parameters on radius of curvature of the obtained dome-shaped part are investigated experimentally. The Taguchi experimental design method is employed to study various parameters, which significantly affect the amount of radius of curvature in the laser forming of a doubly curved dome-shaped part. The length of irradiating line, number of irradiating lines, laser beam diameter, laser output power, and scanning speed are considered in the evaluations. AL27(35) Taguchi standard orthogonal array is chosen for the design of experiments. The level of importance of the process parameters on the radius of curvature is determined by using analysis of variance. The optimum process parameters combination is obtained by using the analysis of signal-to-noise (S:N) ratios. The variation of radius of curvature of the obtained dome-shaped part with process parameters is mathematically modeled by using the regression analysis method and a closed form equation is derived. The results show the length of irradiating line, number of irradiating lines, laser output power, laser beam diameter, and scanning speed have the most importance effects on the radius of curvature of the obtained dome-shaped part, respectively. Also, it is concluded from results that there is an optimum length for irradiating line that leads to minimum radius of curvature of obtained dome-shaped part.

1. Introduction
Laser forming is a flexible forming process that needs no hard tooling or external forces. Laser forming is used in many sheet metal processes such as bending and spatial forming of plates and alignment of metallic and nonmetallic components. in laser forming, a sheet is irradiated with a defocused laser beam, thereby inducing rapid localized heating followed by cooling as the laser beam moves on to an adjacent area. in the heating stage, if the thermal strains in the irradiated zone exceed the elastic strains, then these strains are converted into plastic compressive strains. in the cooling stage, the sheet will undergo shrinkage and therefore a complex shape of the workpiece can be formed. Many of researches in the field of laser forming have been focused on two-dimensional (2D) laser forming. However, in order to advance the process further to industrial applications, it is necessary to study and digest three-dimensional (3D) laser forming. 3D laser forming is very complicated in comparison with 2D studies, and therefore few researches have been reported in this field. Ueda et al. (1994) proposed a strain-based method for line heating of a plate. Using finite element method, they computed strains and decomposed the strains into the in-plane and bending strains. Then the regions with large in-plane strains were selected as heating zones and heating direction was selected normal to the minimum principal strain. Jang and Moon (1998) developed an algorithm to determine heating lines for plate forming by line heating method. They first calculated the lines of curvature of a prescribed surface and evaluated the points of extreme principal curvatures along the lines. They then classified and grouped those points based on their principal directions and their distances. They obtained heating lines by linear regressions on the grouped points. Yu et al. (2000) presented algorithms for optimal development of a smooth continuous curved surface into a planar shape. The development process is modeled by in-plane strain from the curved surface to its planar development. Ishiyama and Tango (2000) proposed a method to determine heating paths where the heating lines for bending strain and in-plane strain are independently calculated by using two procedures:contour heating method andconversion algorithm to the orthogonal compressive inherent strain.
Hennige (2000) investigated the differences in the forming behavior of sheet metal parts using straight and curved irradiations. Also, using radial and circular laser scan paths, he produced a dome-shaped plate from a circular blank. Shin and Lee (2002) proposed a nondimensional relationship between input parameters and final deformation during line heating process by using the flame heat. Kim (2009) developed a method which is reliable and easy to apply, to solve the inverse problem in the laser forming process. They proposed a distance-based method and an angle-based method to generate an irradiation path strategy and to obtain desired angles at each irradiation path. Liu et al. (2004) suggested an optimal process planning strategy to determine scanning paths and heating condition for laser forming of general doubly curved shapes. They also studied two distinctive types of doubly curved surfaces, pillow, and saddle shape, and validated their proposed methodology by experiments. Zhang et al. (2007) simulated laser forming of a plate with a B-spline curve path. They also investigated various temperature fields, displacement fields, and stress and strain fields. Their results showed that peak temperatures of the upper surface and the warped curvature increase when the path curvature increases. Kim and Na (2003) proposed a new method for 3D laser forming of sheet metals. Their method used geometrical information rather than a complicated stress-strain analysis. Using this method, they showed that total calculation time is reduced considerably while provided enhanced accuracy. Chakraborty et al. (2012) used a combination of radial and circular laser scan paths and produced a bowl-shaped surface. Also, they investigated the effects of various process parameters, such as laser spot diameter, laser power, and scan speed, on the in-plane and out-of-plane forming of stainless steel circular blanks for various circular and radial scan schemes. In the all of previous researches that were mentioned earlier, there was not suggested a comprehensive and easy method for production of doubly curved dome-shaped part. Some of them have limited capabilities in the production of doubly curved shapes and the other have very much complexities in the prediction of heating lines for proposing an irradiating scheme. Also, heating lines that are suggested with those methods are very complex curves and some of them are difficult to use in a CNC controller.