跟simple harmonic oscillator有关的一题

╰☆ヾo.海x

An electrical circuit containing an inductor ($L$), a resistor ($R$) and a capacitor ($C$) with a time-varying applied voltage acts as a damped, driven harmonic oscillator, with the equation:
\[
L \frac{d^2 q}{d t^2}+R \frac{d q}{d t}+q / C=V_0 \cos \omega t
\]
(i) Show that the natural frequency, $\omega_0=1 / \sqrt{L C}$
(ii) What role does the resistor $R$ play in the circuit, compared to an oscillator?
(iii) The amplitude for a driven, damped harmonic oscillator is given by:
\[
A=\frac{V_0}{L}\left[(\omega_0^2-\omega^2)^2+\frac{R^2 \omega^2}{L^2}\right]^{-\frac{1}{2}}
\]
For a circuit with $L=22 m H, C=2.2 n F, R=500 \Omega$ and $V_0=3 V$, find the natural frequency, and the amplitude if the driving frequency is 1 kHz. Is this mass (impedance) dominated or stiffness (capacitance) dominated?
第三小问的最后一问：“Is this mass dominated or stiffness dominated?” 这个不会判断。

icesheep

题目就是问你这个阻抗是容抗还是感抗，根据方程就是一个简单的 RLC  串联电路，
阻抗为 \[Z = R + j\omega L + \frac{1}{{j\omega C}}\]
代入数据发现阻抗虚部为负数，因此为容抗。

海盗船长

我也写下吧。。。

那两个概念的定义貌似没有中文对应，上网查到的：
Below resonance, the system is said to be ``stiffness dominated''.
Above resonance, the system is said to be ``mass dominated'' and the resulting displacement approaches zero with increasing frequency.
ccrma.stanford.edu/CCRMA/Courses/152/resonance.html

就是说如果驱动频率高于谐振频率就是mass dominated，低于就是stiffness dominated.

然后就是如何确定谐振频率的问题。

如果认为$q$的最大值最大的时候共振（对应阻尼振子的振幅共振），那么用题目里给的公式使A最大可以得到共振时$$\omega^2=\frac{1}{LC}-\frac{R^2}{2L^2}$$

如果认为$i$的最大值最大的时候共振（对应阻尼振子的速度共振），那么共振的时候就是$$\omega^2=\omega_0^2=\frac{1}{LC}$$

╰☆ヾo.海x

好专业...谢谢楼上2位!!!