Energy Balanced Model of a Jet Interacting With a Brass Player's Lip

In most physical models of brass instruments, the air jet (located at the exciter) is governed by an equation of Bernoulli-type for which basic, unsteady or lossy versions are available. The non-linearity introduced by such model is known to be of utmost importance for wind instruments: it is responsible for self-oscillations when fed by a power source. However, this type of model is unable to make the air flow work at a mobile boundary (lip) as it does not account for any transverse component (the vertical dimension during normal brass playing) in the velocity field. In this sense, it is ill adapted to properly address the power exchanges between the jet and the lips. This paper addresses the following twofold issue. The first issue is the derivation of an air flow model in the unsteady case that restores the fundamental property of passivity at boundaries. The second issue is the test of its relevance in self-oscillating contexts, based on passive-guaranteed simulations of a simplified complete instrument. To make the air flow model as simple as possible, its derivation relies on standard Bernoulli assumptions (irrotational, incompressible flow without loss) except for the boundary conditions at the wall. It is derived in two steps. First, we solve the 2D velocity and pressure fields of the Euler equation for boundary conditions that are adapted to a moving lip. Second, we derive a macroscopic model based on averaged velocities and pressures. This yields a finite dimensional differential system which proves to be equivalent to the original one, in the sense that both the velocity and the pressure fields can be recovered from the macroscopic variables. Moreover, it preserves the power balance of the original physical system. Then, a simplified model of an instrument is built by connecting this system to a lip (mass-spring-damper) and to a conservative straight pipe with an ideally dissipative termination. Passivity is fulfilled and the complete power balance is explicitly encoded by recasting the model into the “Port-Hamiltonian formulation”. A numerical scheme that preserves passivity is proposed to simulate the complete system supplied by an ideal air pressure generator. Finally, results are compared with those based on the standard Bernoulli equation.