# 谐振子

## 维基百科，自由的百科全书

${\displaystyle F=-kx\,}$

## 简谐振子

${\displaystyle F=-kx\,}$

${\displaystyle F=ma=-kx\,}$

${\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2))}=-kx}$

${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2))}+{\omega _{0))^{2}x=0}$

${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2))}={\ddot {x))={\frac {\mathrm {d} {\dot {x))}{\mathrm {d} t)){\frac {\mathrm {d} x}{\mathrm {d} x))={\frac {\mathrm {d} {\dot {x))}{\mathrm {d} x)){\frac {\mathrm {d} x}{\mathrm {d} t))={\frac {\mathrm {d} {\dot {x))}{\mathrm {d} x)){\dot {x))}$

${\displaystyle {\frac {\mathrm {d} {\dot {x))}{\mathrm {d} x)){\dot {x))+{\omega _{0))^{2}x=0}$
${\displaystyle \mathrm {d} {\dot {x))\cdot {\dot {x))+{\omega _{0))^{2}x\cdot \mathrm {d} x=0}$

${\displaystyle {\dot {x))^{2}+{\omega _{0))^{2}x^{2}=K}$

${\displaystyle {\dot {x))^{2}=A^{2}{\omega _{0))^{2}-{\omega _{0))^{2}x^{2))$
${\displaystyle {\dot {x))=\pm {\omega _{0)){\sqrt {A^{2}-x^{2))))$
${\displaystyle {\frac {\mathrm {d} x}{\pm {\sqrt {A^{2}-x^{2))))}={\omega _{0))\mathrm {d} t}$

${\displaystyle {\begin{cases}\arcsin {\frac {x}{A))=\omega _{0}t+\phi \\\arccos {\frac {x}{A))=\omega _{0}t+\phi \end{cases))}$

${\displaystyle x=A\cos {(\omega _{0}t+\phi )}\,}$

${\displaystyle x=A\sin {(\omega _{0}t+\phi )}\,}$

${\displaystyle x=A\sin {\omega _{0}t}+B\cos {\omega _{0}t}\,}$

${\displaystyle f={\frac {\omega _{0)){2\pi ))}$

${\displaystyle T={\frac {1}{2))m\left({\frac {\mathrm {d} x}{\mathrm {d} t))\right)^{2}={\frac {1}{2))kA^{2}\sin ^{2}(\omega _{0}t+\phi )}$.

${\displaystyle U={\frac {1}{2))kx^{2}={\frac {1}{2))kA^{2}\cos ^{2}(\omega _{0}t+\phi )}$

${\displaystyle E={\frac {1}{2))kA^{2))$

## 受驱谐振子

${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2))}+{\omega _{0))^{2}x=A_{0}\cos(\omega t)}$

## 阻尼谐振子

${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2))}+{\frac {b}{m)){\frac {\mathrm {d} x}{\mathrm {d} t))+{\omega _{0))^{2}x=0}$

${\displaystyle \omega _{1}={\sqrt {\omega _{0}^{2}-R_{m}^{2))))$

${\displaystyle R_{m}={\frac {b}{2m))}$

## 受驱阻尼振子

${\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2))}+r{\frac {\mathrm {d} x}{\mathrm {d} t))+kx=F_{0}\cos(\omega t)}$

${\displaystyle x(t)={\frac {F_{0)){Z_{m}\omega ))\sin(\omega t-\phi )}$

${\displaystyle Z_{m}={\sqrt {r^{2}+\left(\omega m-{\frac {k}{\omega ))\right)^{2))))$

${\displaystyle Z=r+i\left(\omega m-{\frac {k}{\omega ))\right)}$

${\displaystyle \phi =\arctan \left({\frac {\omega m-{\frac {k}{\omega ))}{r))\right)}$

${\displaystyle {\omega }_{r}={\sqrt ((\frac {k}{m))-2\left({\frac {r}{2m))\right)^{2))))$

## 完整数学描述

${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2))}+{\frac {b}{m)){\frac {\mathrm {d} x}{\mathrm {d} t))+{\omega _{0))^{2}x=A_{0}\cos(\omega t)}$

${\displaystyle f={\frac {\omega }{2\pi ))}$

### 重要项

• 振幅：偏离平衡点的最大的位移量。
• 周期：系统完成一个振荡循环所需的时间，为频率的倒数。
• 频率：单位时间内系统执行的循环总数量（通常以赫兹 = 1/秒为量度）。
• 角频率${\displaystyle \omega =2\pi f}$
• 相位：系统完成了循环的多少（开始时，系统的相位为零；完成了循环的一半时，系统的相位为${\displaystyle \pi }$）。
• 初始条件t = 0时系统的状态。