Nonlinear vibration response of expanded polymer foam cushion material

The protection of shock and vibration sensitive products from such inputs in the distribution environment is often handled through the use of various cushion materials. Typical cushion materials are made from closed cell expanded polymer foams such as polyethylene, polypropylene, and polystyrene. Textbooks on protective packaging design and packaging dynamics suggest linearized models to predict shock and vibration response, however these materials are highly nonlinear in particular regions of their deflection. These materials are also viscoelastic in nature meaning their behaviour at any instant in time is dependent on their load history. The gap between available models and actual material behaviour is addressed by many cushion designers through the use of cushion curves to predict shock response or compare the shock mitigating performance of different materials. Industry accepted test standards such as ASTM D1596 and ASTM D4168 are used in creating these curves, however significant laboratory time is required to collect the necessary data. Few tools exist for predicting the vibration response of a given mass-loaded cushion material and there are currently no industry accepted standards for creating these tools. This paper provides an overview of some techniques used in determining the frequency response of mass-loaded cushion materials and illustrates their limitations in capturing the nonlinear vibration response. Numerical data from a nonlinear, viscoelastic model of a mass-loaded cushion material is used to demonstrate the frequency response to a sinusoidal base excitation. The model is based on a single degree of freedom dynamic system exposed to linear unidirectional motion. This model includes polynomial type strain-based stiffness, and the viscoelastic component is represented with a hereditary model using a convolution integral formulation. A numerical parameter study of the complete model is conducted to illustrate the effect of different model parameters on frequency response. Finally, an experimental technique is discussed that enables capturing the nonlinear frequency response of a cushion material. Some laboratory data will be illustrated to demonstrate the nonlinear frequency response predicted by the numerical simulation.