Development of a Computational Model Calculating Stress Distribution of a Coil and Coil Slippage in a Coiling Process

DoctorThis study presents a computational model to define several physical phenomena when thin stainless steel strip is coiled, and analyzes the systems with parametric analysis when design variable and coiling condition are changed. In the strip coiling process, the sleeve is a hollow cylinder that is mounted between the coil strip and mandrel to maintain uniform coil shape. But its deformation behavior has not been fully clarified. A simple finite element (FE) model to calculate the stress distribution in the sleeve and coil is presented basically when stainless steel with 1~3 mm thickness is being wound around the sleeve. The FE model is developed by extending the previous model by adding the sleeve between the mandrel and coil, and by modifying the boundary and interaction conditions. After establishing the FE model for all coiling processes, the analytical model of the steady-state process is established to calculate the stress distribution. The analytical model is compared with the FE model to validate these models. Radial & hoop stresses on the sleeve or coil is calculated about 400 layers lamination of the strip. von-Mises stress is compared to yield stress of the sleeve to determine whether the sleeve undergoes plastic deformation or not. To wind coil safely and efficiently, coiling tension, σ, and coiling speed, v, are important factors. Their relationships with the outcome have not been clearly established. Especially, non-linear behavior among contact layers has been treated as a big task. Based on the FE element model of the coiling process, this study suggests that reaction force, FR, on the center of the sleeve is a key variable in the dynamics of the coiling process to understand slip and stick (S&S) phenomenon among contact layers. The FR is investigated by frequency analysis and parametric analysis. Direction cosine, θload, of FR vector is investigated by simulating the FE models and by comparing dynamic responses to a 1-DOF mass-spring system with the S&S phenomenon. Especially, when σ is sufficient, θload is applied at ~0° horizontally to the center of the sleeve. As σ increases, unsteady vibrations of FR also decrease. However, when σ decreases, the natural mode vibration of the strip predominates. This study suggests the coiling status criterion by checking θload vector in the time-frequency domain depending on σ and v, and by separating frequencies of the θload vector into the natural mode characteristics of the strip and the slip phenomenon between the sleeve and strip. Based on the criterion, this study also suggests a profile of σ vs. number of layers of coil which considers vibration data. To avoid the S&S phenomenon among contact layers, the FR is selected to represent the coiling status. In parametric analysis, FR is arranged with different σ and the outer radius of the coil, ro. Results of FR are transferred in the frequency domain by the Fourier transform and by the magnitude-squared coherences (MSC). Even though the area of coherence, Areac, of σ shows irregular distributions partially, high σ is required to reduce coiling vibrations of FR,x as ro decreases. Especially, σ is very sensitive to coiling status when ro is small. By using the suggested σ profile, the slip is infrequent. Therefore, coil quality can be guaranteed. The suggested σ profile reduces the radial stress on the sleeve, so that maintenance costs can be minimized by preventing the sleeve and coil from dents. To investigate the coiling system that uses the belt wrapper, the FE coiling model and the analytical model are proposed; the belt wrapper presses the coil to fix on the outermost surface of the sleeve. The belt wrapper is quantified how the slip mechanism between the sleeve and coil are affected by coiling operating conditions. When the modulus of elasticity, Ebelt, of the belt wrapper is higher than the critical value, rate of pressure between the outermost surface of the belt wrapper and the innermost surface of the sleeve decreases linearly as Ebelt is increased. Slip between the coil and sleeve happens rapidly when radial stress on the sleeve caused by strip tension, T, is higher than the radial stress on the sleeve. Stress distributions of the sleeve and coil are the same or lower when the coil did not undergo plastic deformation in comparison to the model with the grooved joint. Mathematical model of an un-coiling idle coiling (UCIC) process has been used to estimate the strip tension. The model is combined with an electrical control system to rotate each rotor exactly. Lastly, this study suggests an advanced mathematical model of the UCIC process based on the previous model for winding a web. The suggested model consists of the equations of motion and the state variable equations; geometrical variations including the radius of the coil, r, the eccentricity of the coil, lecc, and the effective strip tension, Teff, with slip behavior among contact layers are renewed automatically in numerical analysis. Especially, Teff shows similar to the strip tension profile compared to the previous tension model when two tension models are validated with a FE model of the UCIC process. Moreover, the planar motions are defined in the equations of the motion to check how T and the dynamic responses of the un-coiler, pinch rollers, and coiler affect to each other. The planar motions of each rotor have translational motions by applying spring stiffness, kx, and, ky, on the center of the rotor that correspond to the elastic deformation of the bearing and shaft. By using the suggested model, T and unusual vibrations that are caused by improper coiling conditions are predicted