Modal Synthesis for Vibrating Objects ∗

When a solid object is struck, scraped, or engages in other external interactions, the forces at the contact point causes deformations to propagate through the body, causing its outer surfaces to vibrate and emit sound waves. Examples of musical instruments utilizing solid objects like this are the marimba, the xylophone, and bells. The sounds made by objects like this are important for interacting with our environment because they provide useful information about the physical attributes of the object, its environment, and the contact events, including the force (or energy) of the impact, the material composition of the object, the shape and size, the place of impact on the object, and finally the location and environment of the object. In order to create the sounds of objects like this in an interactive digital environment, such as a video game or a simulation, we need real-time synthesis, as we do not know the stimulus of the (virtual) objects before they occur, and sustained intimate user interaction like touching and scraping an object needs a continuously parametrizable sound. A good physically motivated synthesis model for objects like this is modal synthesis (Wawrzynek, 1989; Gaver, 1993; Morrison & Adrien, 1993; Cook, 1996; Doel & Pai, 1996; Doel, Kry, & Pai, 2001; O’Brien, Chen, & Gatchalian, 2002; Doel, Pai, Adam, Kortchmar, & Pichora-Fuller, 2002), where a vibrating object is modeled by a bank of damped harmonic oscillators which are excited by an external stimulus. The frequencies and dampings of the oscillators are determined by the geometry and material properties (such as elasticity) of the object and the coupling gains are determined by the location of the force applied to the object. The modal synthesis model is physically well motivated, as the linear partial differential equation for a vibrating system, with appropriate boundary conditions, has as solutions a superposition of vibration modes. See Fig. 1 for an illustration. Modal synthesis can also be used to model other types of physical systems which can be modeled by excitations acting on resonances, such as car engines, rumbling sounds, or virtual musical instruments. For musical instruments with a harmonic spectrum modal synthesis can be used, but it is computationally quite demanding because of the large number of modes needed. Waveguide models (Smith, 1992; Cook, 2003) are in most cases much more efficient for these types of sounds. The sound made by a modal model can be computed very efficiently with an O(N) algorithm (Gaver, 1993; Doel & Pai, 1998; Doel, 1998) for a model of N modes, as described below.