# 諧振子

（重定向自簡諧振子

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

## 簡諧振子

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## 受驅諧振子

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

## 阻尼諧振子

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

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

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

## 受驅阻尼振子

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

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

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

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

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

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

## 完整數學描述

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

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

### 重要項

• 振幅：偏離平衡點的最大的位移量。
• 週期：系統完成一個振盪循環所需的時間，為頻率的倒數。
• 頻率：單位時間內系統執行的循環總數量（通常以赫茲 = 1/秒為量度）。
• 角頻率${\displaystyle \omega =2\pi f}$
• 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${\displaystyle \pi }$ ）。
• 初始條件t = 0时系统的状态。