声学基础课件许肖梅fundamentalsofacoustics07-5

1、Analogies Between Mechanical and Electrical Systems,We can develop a deeper understanding of what we have said about the harmonic oscillator by drawing parallels with simple electric circuit. These analogies can help our study of more complex vibrating systems, and we will eventually extend them to

2、apply in acoustic problems as well.,Many vibrating systems are mathematically equivalent to corresponding electrical systems.,We have already studied resonance in a series circuit, much of the preceding discussion should seem quite familiar. From the analogies you could have predicted that the natur

3、al frequency,D=1/Cm , Cm is called mechanical compliance,2. In a damping of the oscillations,the differential equation for the motion becomes :,Consider a simple series electrical circuit containing inductance L, resistance R, and capacitance C, driven by sinusoidal voltage V conwt, as suggested in

4、Fig. a,Fig. a,Equivalent series systems,The differential equation for the charge q is :,The differential equation for the current I=d q /d t is :,We see that the electrical circuit of Fig. a is the mathematical analog of the damped harmonic oscillator of Fig. b,Fig. b,The current I in the electrical

5、 system is equivalent to the speed v in the mechanical system.,The charge Q is equivalent to the displacement x, and the two system have similar forms, with the mechanical resistance Rm analogous to the electrical resistance R, the mass m analogous to the electrical inductance L, and the spring cons

6、tant (or compliance Cm) analogous to the electrical capacitance C.,The Mechanical System,The Electrical System,The elements in the electrical system Fig. a are said to be in series because they experience the same current. Similarly, the elements in the mechanical system Fig. b can be represented by

7、 the series circuit of Fig. b : they experience the same displacement and, therefore, the same speed.,If a simple mechanical oscillator is driven by a sinusoidal force applied to the normally fixed end of the spring as suggested by Fig. c, then the mass and the spring experience the same force and t

8、his combination represented by a parallel circuit, as shown in Fig. d.,Fig. c,For the spring :,For the m :,Substituting v for x,For the case of a sinusoidal driving force,Where the Z1 and Z2 are :,Obtain the speed v1 :,Which has the same form as a series electrical system.,The speed of the driven en

9、d of the spring is equivalent to the current entering the parallel circuit, and the speed v2 of the mass is equivalent to the current flowing through the inductor,Equivalent series-parallel systems,The elements in the mechanical system are said to be in series when they experience the same displacement, The