Appl. Sci. 2022 , 12 , 1684
2of 15
corrugated board, and could prove that the much more easily evaluated analytical model reaches the same accuracy as the numerical approach. The most typical product made of corrugated board are boxes for storage and trans- portation. Therefore, the load-bearing capability of those boxes is also of interest. Ref. [8] provides a review of the research on the load-bearing capability of corrugated board boxes depending of various influence factors (e.g., moisture, holes, or inserts). Recently, a series of articles [9–11] systematically analyzed the influence of various modifications in the basic structure of a standard flap box, namely, the influence of holes, perforations, and offset in the height of the flaps, respectively. Combining numerical finite-element calculations with lab tests, those articles derived improved analytical formulas for estimating the compressive strength of such modified boxes compared to classical and widely used formulas. Produc- tion defects such as warping of the board see less research activity. As in the examples above, often the properties of error-free manufactured corrugated board are investigated. Production defects are nonetheless a practical problem: corrugated board that exceeds a certain limit of warp can no longer be reliably processed and must therefore be considered as waste. Ref. [12] describes practical optimizations to reduce warp in production, but does not address the formation of warp within the corrugated board. This article acts as a first step of a more theory-based approach to process control by providing a mathematical model for the description of warp. In order to reduce waste resulting from the production of warped corrugated board, better process control is necessary. This, in turn, requires the possibility of measuring warp in an automated and cost-efficient way; otherwise, accepting a certain amount of warp (and, consequently, waste) can simply be the more economic decision for running a production plant. A possible solution for this problem is to use a low number of cheaply available spot displacement sensors (usually based on laser triangulation) to measure the height at discrete points distributed sparsely over the surface of the corrugated board, and interpolate between those points. This interpolation should represent the form of (warped) corrugated board as closely as possible with a minimum of measured points. Therefore, this article develops a model describing the surface form of warped corrugated board. The knowledge about the physical properties of corrugated board embedded in this model reduces the degrees of freedom which need to be determined from measured data, and, as a consequence, the number of sensors required. This allows automated, cost-efficient, and robust (ignoring small-scale surface defects) measurement of warp in corrugated board manufacturing, providing a previously unattained level of information about the warping of the produced board. In a future step, this information could be used for real-time process monitoring and control. The model proposed in this article expresses warp as a result of internal stress using classical Kirchhoff plate theory. Thereby, the model incorporates knowledge of the physical causes of warp which helps to reduce the number of measure points required for interpo- lation. In order to maximize the practical utility of the model for analyzing warp in the production process and ultimately provide the necessary measurement tools to dynamically adjust production parameters to reduce warp, no knowledge of the mechanical parameters of the corrugated board is assumed, since those parameters can usually only be obtained by destructive testing. Instead, the number of parameters is reduced as much as possible, allowing reconstruction of the necessary information from the measured data. This model also helps to formalize the concept of warp. Warp is intuitively described as the general trend of the surface over a larger area (as opposed to local fluctuations such as dents or the washboarding effect, which, while also important quality metrics, need to be treated differently in production), a definition which is too vague to actually work with. Figure 1 illustrates the difference between large-scale warp and small-scale surface defects. This means that, even if the exact surface of a corrugated board would be known, the warp would still not be easily quantified. In the corrugated board industry, cumulative values derived from the maximal deviation from an ideal plane are currently routinely used to quantify warp. Ref. [13] calculates the direct quotient between the maximal deflection and
Made with FlippingBook - Online magazine maker