Modal sound synthesis has been used to create realistic sounds from rigid-body objects, but requires accurate real-world material parameters. These material parameters can be estimated from recorded sounds of an impacted object, but existing methods do not account for the additional damping attributed to how the object is held. We present a novel technique for estimating the damping parameters of materials from recorded impact sounds that separates the effects of damping due to the way the struck object is supported. We use a generative model to represent the combined effects of material damping, support damping, and sampling inaccuracies, then use maximum likelihood estimation to fit a damping model to recorded data. This technique simplifies the recording process, only requires audio as input without knowing the precise object geometry or the exact hit location, and can use multiple recorded impact sounds to improve accuracy. We validate the technique with a comprehensive analysis of a synthetic dataset and a perceptual material identification user study.