## 1 Introduction
In [42], the authors studied the damping properties of a 3D-printed photopolymer with Kelvin model due to impact loading. The standardized method to obtain the cushioning characteristics of a specific material was described in the American Society for Testing and Materials [43; 44]. In addition to shock events, vibration inputs to items also can be critical if their frequencies are the same or similar to the natural frequencies of the product or packaging. Cushioning must be offered protection from these inputs as well. Therefore, the cushioning material must attenuate vibration inputs at those frequencies or similar ones. However, cushion curves cannot be applied in the prediction of the characterization of the frequency response of a given material, because they represent only static loading vs. shock experienced as the cushioning is impacted [45]. The packaging industry does not have a standard method for testing the designed frequency response of a cushioning material [45]. Developing a transmissibility curve is one of the possible ways to determine the frequency response of a given material [45]. Such a curve can be generated by using a sine sweep or random vibration inputs. The curve is limited to the sample parameter [46]. Several researchers have determined vibration transmissibility curves for different materials, but only in normal temperatures [4; 47; 48; 49; 50; 51; 52]. An application of a vibration-attenuation plot during the package-design process was shown by [PERSON] et al. [52]. [PERSON] et al. [53], [PERSON] et al. [54], and [PERSON] [45] developed different techniques to study and model the nonlinear effect of cushioning on the estimation of vibration transmissibility, but without investigating temperature changes. The lack of knowledge about the effect of temperature changes on dynamic packaging properties, including vibration transmissibility, limits these applications in designing protective packaging for products. Despite the fact that mechanical properties of PE are affected by temperature, and packaged products encounter different temperature conditions (in extreme cases from \(-20\)\({}^{\circ}\)C to \(50\)\({}^{\circ}\)C, according to [PERSON] et al. [33]) in the distribution environment, limited published studies are available on the effect of temperature on cushioning designs. [PERSON] et al. [55], [PERSON] [56], and [PERSON] et al. [33] studied the effect of temperature on the shock properties. [PERSON] et al. [55] also observed the effect of temperature on vibration transmissibility, but it was only applied during a preconditioning period, and was not used during testing. Furthermore, the study used random vibration signals, which are not a clear harmonic-generated motion, and it was only applied to normal PE foam and EPS (extruded polystyrene). However, we could not find any published laboratory cushioning research that measured and analyzed the vibration transmissibility and damping ratios of XPE for package cushioning using different temperature and static-load conditions, or for PE material that applied temperature conditions during the entire measurement. Our study presents new measured and analyzed data that can help packaging engineers gain a better understanding of the characteristics of these materials, and to design appropriate protective packaging for sensitive goods. The main goal of our study is to present the critical frequency bands and possible damping ratios of PE and XPE cushion material for cushioning. This paper also discusses the effect of temperature changes during vibration circumstances and gives relevant information for varied static loads. The data from this research can be compared to previous research that was performed for shock properties of PE materials. This new data can be a useful technical support for engineers in packaging design. Our study provides more accurate details about the vibration transmissibility of plastic foams at different temperatures, and clearly shows the effect of temperature.