強制振動

解

$ \ddot{x}+\omega^2x=f(t)

$ \iff(D+i\omega)(D-i\omega)x=f(t)

$ \iff D(e^{i\omega t}(D-i\omega)x)=e^{i\omega t}f(t)=:F'(t)

$ \iff(D-i\omega)x=F(t)e^{-i\omega t}+Ae^{-i\omega t}\quad\text{.for }\exist A\in\Complex

$ \iff D(e^{-i\omega t}x)=F(t)e^{-2i\omega t}+Ae^{-2i\omega t}=:G'(t)+Ae^{-2i\omega t}\quad\text{.for }\exist A\in\Complex

$ \underline{\iff x=G(t)e^{i\omega t}+Ae^{i\omega t}+Be^{-i\omega t}\quad\text{.for }\exist A,B\in\Complex\quad}

これは当然

$ A_f,B_f\in\Complex,\omega_f\in\R\setminus\{0\}

$ \ddot{x}+\omega^2x=f(t)

$ \iff(D+i\omega)(D-i\omega)x=A_fe^{i\omega_ft}+B_fe^{-i\omega_ft}

$ \iff D(e^{i\omega t}(D-i\omega)x)=A_fe^{i(\omega_f+\omega)t}+B_fe^{-i(\omega_f-\omega)t}

$ = D\frac{A_f}{i(\omega_f+\omega)}e^{i(\omega_f+\omega)t}-D\frac{B_f}{i(\omega_f-\omega)}e^{-i(\omega_f-\omega)t}\llbracket\omega_f\neq\omega\rrbracket+B_f\llbracket\omega_f=\omega\rrbracket

$ \iff(D-i\omega)x=\frac{A_f}{i(\omega_f+\omega)}e^{i\omega_ft}-\frac{B_f}{i(\omega_f-\omega)}e^{-i\omega_ft}\llbracket\omega_f\neq\omega\rrbracket+B_fte^{-i\omega_ft}\llbracket\omega_f=\omega\rrbracket+Ce^{-i\omega t}\quad\text{.for }\exist C\in\Complex

$ \iff D(e^{-i\omega t}x)=\frac{A_f}{i(\omega_f+\omega)}e^{i(\omega_f-\omega)t}-\frac{B_f}{i(\omega_f-\omega)}e^{-i(\omega_f+\omega)t}\llbracket\omega_f\neq\omega\rrbracket+B_fte^{-2i\omega t}\llbracket\omega_f=\omega\rrbracket+Ce^{-2i\omega_ft}\quad\text{.for }\exist C\in\Complex

$ =-D\frac{A_f}{\omega_f^2-\omega^2}e^{i(\omega_f-\omega)t}\llbracket\omega_f\neq\omega\rrbracket-D\frac{B_f}{\omega_f^2-\omega^2}e^{-i(\omega_f+\omega)t}\llbracket\omega_f\neq\omega\rrbracket+\frac{A_f}{2i\omega_f}\llbracket\omega_f=\omega\rrbracket+B_fte^{-2i\omega_ft}\llbracket\omega_f=\omega\rrbracket+Ce^{-2i\omega t}\quad\text{.for }\exist C\in\Complex

$ \iff x=-\frac{1}{\omega_f^2-\omega^2}(A_fe^{i\omega_ft}+B_fe^{-i\omega_ft})\llbracket\omega_f\neq\omega\rrbracket+\left(\frac{A_f}{2i\omega_f}te^{i\omega_ft}+B_f\frac{t-\frac1{-2i\omega_f}}{-2i\omega_f}e^{-i\omega_ft}\right)\llbracket\omega_f=\omega\rrbracket+Pe^{i\omega t}+Qe^{-i\omega t}\quad\text{.for }\exist P,Q\in\Complex

$ =-\frac{f(t)}{\omega_f^2-\omega^2}\llbracket\omega_f\neq\omega\rrbracket+\left(\frac{A_f}{2i\omega_f}te^{i\omega_ft}-\frac{B_f}{2i\omega_f}te^{-i\omega_ft}\right)\llbracket\omega_f=\omega\rrbracket+\frac{B_f}{2\omega_f^2}e^{-i\omega_ft}\llbracket\omega_f=\omega\rrbracket+Pe^{i\omega t}+Qe^{-i\omega t}\quad\text{.for }\exist P,Q\in\Complex

$ =-\frac{f(t)}{\omega_f^2-\omega^2}\llbracket\omega_f\neq\omega\rrbracket+\frac{it}{2\omega_f}\left(-A_fe^{i\omega_ft}+B_fe^{-i\omega_ft}\right)\llbracket\omega_f=\omega\rrbracket+Ae^{i\omega t}+Be^{-i\omega t}\quad\text{.for }\exist A,B\in\Complex

$ =-\frac{f(t)}{\omega_f^2-\omega^2}\llbracket\omega_f\neq\omega\rrbracket-\frac{f(t)-2B_fe^{-i\omega_ft}}{2\omega_f}it\llbracket\omega_f=\omega\rrbracket+Ae^{i\omega t}+Be^{-i\omega t}\quad\text{.for }\exist A,B\in\Complex

$ \underline{\iff x(t)=-\frac{f(t)}{\omega_f^2-\omega^2}\llbracket\omega_f\neq\omega\rrbracket-\frac{f(t)-2B_fe^{-i\omega_ft}}{2\omega_f}it\llbracket\omega_f=\omega\rrbracket+Ae^{i\omega t}+Be^{-i\omega t}\quad\text{.for }\exist A,B\in\Complex\quad}

$ \iff \Im A_f\Re e^{i\omega_ft}+\Re A_f\Im e^{i\omega_ft}+\Im B_f\Re e^{-i\omega_ft}+\Re B_f\Im e^{-i\omega_ft}=0

$ \iff\Im(A_f+B_f)\cos\omega_f t+\Re(A_f-B_f)\sin\omega_f t=0

$ \Im(-A_fe^{i\omega_ft}+B_fe^{-i\omega_ft})=\Im(-A_f+B_f)\cos\omega_f t+\Re(-A_f-B_f)\sin\omega_f t

まとめられはするけど、複雑だ

$ \Re f(t)=\Re(A_f+B_f)\cos\omega_ft+\Im(B_f-A_f)\sin\omega_ft

$ f(t)=\frac{A_f'-iB_f'}{2}e^{i\omega_ft}+\frac{A_f'+iB_f'}{2}e^{-i\omega_ft}

$ \implies-A_fe^{i\omega_ft}+B_fe^{-i\omega_ft}=-\frac{A_f'-iB_f'}{2}e^{i\omega_ft}+\frac{A_f'+iB_f'}{2}e^{-i\omega_ft}

$ =-iA_f'\sin\omega_ft+iB_f'\cos\omega_ft

$ \therefore x(t)=-\frac{f(t)}{\omega_f^2-\omega^2}\llbracket\omega_f\neq\omega\rrbracket+\frac{A_f'\sin\omega_ft-B_f'\cos\omega_ft}{2\omega_f}t\llbracket\omega_f=\omega\rrbracket+Ae^{i\omega t}+Be^{-i\omega t}\quad\text{.for }\exist A,B\in\Complex

$ f'(t)=i\omega_fA_fe^{i\omega_ft}-i\omega_fB_fe^{-i\omega_ft}

$ =-i\omega_f(-A_fe^{i\omega_ft}+B_fe^{-i\omega_ft})

$ \underline{\therefore x(t)=-\frac{f(t)}{\omega_f^2-\omega^2}\llbracket\omega_f\neq\omega\rrbracket-\frac{f'(t)}{2\omega_f^2}t\llbracket\omega_f=\omega\rrbracket+Ae^{i\omega t}+Be^{-i\omega t}\quad\text{.for }\exist A,B\in\Complex\quad}

わかること

$ x(t)=\underbrace{\frac{f(t)}{\omega^2-\omega_f^2}\llbracket\omega_f\neq\omega\rrbracket}_\text{共鳴しないときの外乱}-\underbrace{\frac{f'(t)}{2\omega_f^2}t\llbracket\omega_f=\omega\rrbracket}_\text{共鳴するときの外乱}+\underbrace{Ae^{i\omega t}+Be^{-i\omega t}}_\text{系の固有振動}\quad\text{.for }\exist A,B\in\Complex

共鳴すると、外乱によって振幅が線型に広がる非定常運動になる

外乱に複数の角振動数が混じっていた場合

それぞれの外乱の角振動数を重ね合わせればいいだけ

$ \ddot{x}+\omega^2 x=\sum_i f_i(t):=\sum_i A_{fi}e^{i\omega_{fi}t}+B_{fi}e^{-i\omega_{fi}t}

$ \implies x(t)=\sum_i\frac{f_i(t)}{\omega^2-\omega_{fi}^2}\llbracket\omega_{fi}\neq\omega\rrbracket-\sum_i\frac{f_i'(t)}{2\omega_{fi}^2}t\llbracket\omega_{fi}=\omega\rrbracket+Ae^{i\omega t}+Be^{-i\omega t}\quad\text{.for }\exist A,B\in\Complex