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Estimating parameters of a silencer by solving Helmholtz equation using finite elements method

The transmission loss (TL), noise reduction (NR) and insertion loss (IL) coefficients are the most widely used to describe the performance of mufflers and silencers as acoustic systems. The possibility to estimate these coefficients as a function of frequency for a given geometry is of utmost importance in modern acoustic use. Geometry parameters of the suggested silencer are first imported in a set of Matlab functions to verify the model using 1-D acoustic transmission line theory. The TL and transfer coefficients as a function of frequency are estimated using the plane wave theory propagation for restricted frequency bandwidth (f<fc) and setting the appropriate acoustic impedance at outlet radiation surface. This model has shown good characteristics at lower frequencies, but the appearance of higher modes of acoustic waves is neglected in this model. Due to restrictions of the described model for higher frequencies, MATLAB PDE Toolbox has been used to solve the problem using finite elements method. The linear plane wave equation for pressure field is assumed and then transformed to the well-known Helmholtz equation. This equation is then solved in a subdomain with appropriate boundary conditions. On metal faces of the system it is assumed that normal component of velocity equals zero. On the inlet surface it is assumed that reflecting waves are totally absorbed and only the input wave exists (produced by a vibrating loudspeaker membrane) at this surface. On the outlet surface Neumann radiation condition is assumed because radiation impedance at the outlet (its characteristics equivalent) is assumed not to be a constant value, but a frequency dependent complex radiation impedance of flanged or unflanged end of pipe. Relative sound pressure level, velocity distribution and transfer coefficients are shown for different frequencies of input wave. TL is calculated as the ratio of outlet sound power (connected with out-pressure wave) and input sound power (incident pressure wave). Simulations of transfer coefficients for wide frequency bandwidth are done by repeating the calculations for the described silencer model for each individual frequency in for loop. Estimated transfer coefficient in form of ratio of output pressure and total input wave have been compared with measurement results for realized silencer.

silencer parameters; 1-D acoustic transmission line theory; finite elements method; boundary conditions

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