jats:p Materials or structures made of polymers or composites often exhibit non-viscous damping behavior. The non-viscous damping forces that depend on the history velocity can be expressed by the exponential non-viscous damping model. Some polymer or composite materials can be used as damping materials for kinetic energy absorption in the dynamic systems. Vibration isolator and absorber are usually considered for shock absorption. In this study, transfer ratios of vibration isolator and absorber with exponential non-viscous damping system are derived by using the Laplace transform. The dimensionless amplitude of vibration absorber with exponential non-viscous damping is derived too. Compared to viscous damping system, transfer ratio and dimensionless amplitude of exponential non-viscous damping system are influenced by the ratio of the relaxation parameter and natural frequency or the frequency of the external load. With the non-viscous damping material used in vibration control, the ratio is therefore a non-negligible factor which should be considered in analysis. </jats:p>