Free Vibration analysis of circular membrane backed by a cylindrical cavity using Impedance-Mobility Approach

The membrane backed by a cavity is used in many applications like sensors, loudspeaker design and musical acoustics. There are two types of configurations existing in engineering applications such as single membrane cavity and double membrane cavity. The modal parameters of membrane are influenced by the supporting back cavity due to structural-acoustic coupling. In this study, pre-stressed membrane with back cavity has been analyzed by using Impedance Mobility Coupling Method (IMCM). The coupled behavior of the system is expressed in terms of the finite number of uncoupled subsystem modes. The present paper discusses a generalized formulation to calculate modal parameters of coupled system and has demonstrated it for a single membrane cavity problem. Coupling between acoustic and structural subsystem is expressed in terms of a mode coupling coefficient which describes spatial matching of both mode shapes. Acoustic uncoupled subsystem is a circular cavity at one end closed and other end open for the single membrane case. The length of the cavity is more than its diameter, and the cylindrical surface is acoustically rigid. The membrane is thin and is coupled to the closed end of the cavity by clamping its circular edge. Radial tension along the circumference of the thin membrane is applied to provide membrane pre-tension. For the theoretical analysis, the acoustic impedances of all the cavity modes and the structural mobility of all the membrane modes are calculated. The complex amplitudes of the acoustic pressure and vibration velocity are then arranged in a matrix form and modified suitably to prepare as a standard eigenvalue problem. The numerical estimation of the coupled frequencies is carried out by coupling structural and acoustic physics in a Finite Element (FE) environment. In structural case, membrane physics was established and the initial tension to the membrane was provided by giving initial in-plane forces. The membrane was provided with fixed constraints at the edge to replicate fixed constraints at the boundary in case of theoretical analysis. Zero acoustic pressure condition was imposed at one end of acoustic cavity to mimic the open-close boundary condition while remaining surfaces were defined acoustically rigid. Velocity continuity condition was defined as coupling boundary condition. The predicted numerical and theoretical coupled natural frequencies are in good agreement with each other. The proposed methodology is useful to predict the Eigen frequencies of the weakly coupled system