THREE DEGREES OF FREEDOM 61 



The force applied to the mass and resistance is 



Jm = mxz + r^Xi 4.55 



The Acoustical System 



The total volume displacement, in the acoustical system of Fig. 4.3, is 

 the sum of the volume displacement of the inertance M and the volume 

 displacement of the acoustic capacitance Ca- 



Xi = X2 + Xz 4.56 



The total volume displacement is the volume displacement of the vi- 

 brating piston. The vibrating piston is not a part of the acoustical system. 

 It is merely the sound pressure source which produces the sound pressure^. 



Differentiating equation 4.56 with respect to the time, the volume cur- 

 rents are 



Xi = X, +Xs 4.57 



The pressure applied to the capacitance is 



p= ^ 4.58 



Ca 



The pressure applied to the inertance and acoustic resistance is 



p = MXz + taXz 4.59 



A comparison of the coefficients- of equations 4.49, 4.51, 4.54, 4.55, 4.58 

 and 4.59 shows again that resistance, inductance and capacitance is equiva- 

 lent to resistance, mass and compliance in the mechanical system and to 

 resistance, inertance and acoustic capacitance in the acoustical system. 

 A comparison of equations 4.48, 4.53 and 4.57 shows that currents in the 

 electrical system are analogous to velocities in the mechanical system and 

 to volume currents in the acoustical system. 



The equations for the impedance of a parallel electrical circuit are given 

 in all text-books on electrical engineering and will not be repeated. The 

 performance of the mechanical or acoustical system of Fig. 4.3 may be 

 predicted by employing the equations for the electrical circuit of Fig. 4.3. 



4.8. Electrical, Mechanical and Acoustical Systems of Three Degrees 

 of Freedom. — A system of three degrees of freedom is shown in Fig. 4.4. 

 The currents in the different branches of the electrical system may be 

 obtained by using the rules and formulas of the electrical circuit theory. 

 From the differential equations of the electrical, mechanical and acoustical 

 systems it can be shown Z-i, L2, Cei, Cei and Te in the electrical system are 



