Many packaging cushion materials follow the hyperbolic tangent type of constitutive relation, so a hyperbolic tangent stiffness non-linearity was derived to describe the stiffness property of the package. Transport package was modelled as the single degree of freedom (SDOF) of hyperbolic tangent spring–mass–damping (SMD) system. The displacement and velocity responses of the system were obtained by Fokker–Planck–Kolmogorov (FPK) equation. Based on the kinematic equation and integral transforming, the approximate analytic solution of the acceleration response power spectral density (PSD) of the hyperbolic tangent package was established for the first time, and the influences of system non-linearities on the acceleration response PSD were investigated. Validation experiment with a complete packaged computer was carried out to verify the proposed approach. Non-linearity of the stiffness influenced the acceleration response PSD through the effect of parameter γ and statistical moment m_i0. The frequency corresponding to acceleration response peak was determined by the damping correlation coefficient α and parameter γ, which was determined by system characteristic parameter β, frequency parameter ω₀ and statistical moment m_i0. The proposed approach can be used to predict the acceleration response PSD of transport package, and the analysis has important value for packaging optimization design.

中文翻译：

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随机激励下双曲正切封装的加速度谱分析

许多包装缓冲材料遵循双曲正切类型的本构关系，因此推导出双曲正切刚度非线性来描述包装的刚度特性。运输包被建模为双曲正切弹簧-质量-阻尼 (SMD) 系统的单自由度 (SDOF)。系统的位移和速度响应由 Fokker-Planck-Kolmogorov (FPK) 方程获得。基于运动学方程和积分变换，首次建立了双曲正切包加速度响应功率谱密度(PSD)的近似解析解，研究了系统非线性对加速度响应PSD的影响. 使用完整的封装计算机进行验证实验以验证所提出的方法。刚度的非线性通过参数的影响影响加速度响应PSDγ和统计矩m _{i 0}。加速度响应峰值对应的频率由阻尼相关系数α和参数γ确定，γ由系统特征参数β、频率参数ω ₀和统计矩m _{i 0 确定}。该方法可用于预测运输包装的加速度响应PSD，该分析对包装优化设计具有重要价值。