Tarkibga o'tish

Chastotaviy javob tahlili

Masalaning qo‘yilishi

Chastotaviy javob tahlili tashqi kuch vaqt bo‘yicha garmonik o‘zgarganda tizimning turg‘un holatdagi javob amplitudasi va fazasini chastota sohasida baholaydi. Harakat tenglamasi so‘nishsiz va so‘nishli hollarda turlicha ko‘rilgani sababli, ushbu bob avval so‘nishsiz erkin tebranishdan tabiiy modalarni chiqaradi, so‘ng ularni modal bazis sifatida ishlatib so‘nish mavjud bo‘lgandagi garmonik javobni yoyadi.

So‘nishsiz erkin tebranish va tabiiy modalar

So‘nish hisobga olinmasa, harakat tenglamasi quyidagicha bo‘ladi.

\[\begin{equation} M \ddot{U} + K U = 0 \label{eq:2.6.1} \end{equation}\]

Ko‘chishni har bir tabiiy moda bo‘yicha yoyganda,

\[\begin{equation} U = U_j e^{i \omega_j t} \label{eq:2.6.2} \end{equation}\]

Buni \(\eqref{eq:2.6.1}\) tenglamaga qo‘ysak,

\[\begin{equation} K U_j = \omega_j^2 M U_j \label{eq:2.6.3} \end{equation}\]

quyidagi natija olinadi. Bu umumlashtirilgan xos qiymat masalasidir. Uning sonli yechim usullari, jumladan siljitilgan teskari iteratsiya va Lanczos usuli Modal tahlil da ko‘rib chiqiladi. Ushbu bo‘limda keyingi garmonik javob yoyilmasida ishlatiladigan tabiiy chastotalar va tabiiy modalarning xossalari jamlanadi.

Tabiiy chastotalarning haqiqiyligi

Tabiiy chastotaning haqiqiy son ekanini quyidagicha ko‘rsatish mumkin. \(\omega_j^2 = \lambda_j\) deb olib, \(\eqref{eq:2.6.3}\) tenglamaning kompleks qo‘shmasini olsak, \(\eqref{eq:2.6.4}\) tenglama hosil bo‘ladi.

\[\begin{equation} K U_j = \lambda_j M U_j K \overline{UJ} = \overline{\lambda_J} M \overline{U_J} \label{eq:2.6.4} \end{equation}\]

Buni \(\overline{U}_J^T\) ga ko‘paytirsak,

\[\begin{equation} U_j^T K \overline{U}_J = \overline{\lambda}_J U_j^T M \overline{U}_J \overline{U}_J^T K U_j = \lambda_j \overline{U}_J^T M U_j \label{eq:2.6.5} \end{equation}\]

\(\eqref{eq:2.6.5}\) tenglamadan,

\[\begin{equation} 0 = ( \lambda_j - \overline{\lambda_J} ) \overline{U_J}^T M U_j \label{eq:2.6.6} \end{equation}\]

Massa matritsasi simmetrik musbat aniqlangan bo‘lgani uchun, nol bo‘lmagan xos vektor uchun

\[\begin{equation} \overline{U_J} M U_j > 0 \label{eq:2.6.7} \end{equation}\]

Shuning uchun,

\[\begin{equation} \lambda_j =\overline{\lambda_J} \label{eq:2.6.8} \end{equation}\]

Demak, \(\omega_j^2 = \lambda_j\) haqiqiy sondir.

Tabiiy modalarning ortogonalligi va normallashuvi

Ikki xil modani qaraymiz.

\[\begin{equation} K U_i = \lambda_i M U_i K U_j = \lambda_j M U_j \label{eq:2.6.9} \end{equation}\]

Bundan

\[\begin{equation} ( \lambda_i - \lambda_j ) U_j^T M U_i = 0 \label{eq:2.6.10} \end{equation}\]

kelib chiqadi va xos qiymatlar turlicha bo‘lganda,

\[\begin{equation} U_j^T M U_i = 0 \label{eq:2.6.11} \end{equation}\]

Shunday qilib, turli tabiiy modalar massa matritsasiga nisbatan ortogonaldir. Bir xil moda uchun esa \(\eqref{eq:2.6.12}\) tenglamadagidek massa matritsasiga nisbatan normallashtirish keyingi hisob-kitoblarni soddalashtiradi.

\[\begin{equation} U_i^T M U_i = 1 \label{eq:2.6.12} \end{equation}\]

So‘nish mavjud bo‘lgandagi garmonik javob

Endi so‘nish hisobga olingan chastotaviy javob tahlilining formulirovkasi keltiriladi. Qaralayotgan harakat tenglamasi \(\eqref{eq:2.6.13}\) tenglamada berilgan.

\[\begin{equation} M \ddot{U} + C \dot{U} + K U = F \label{eq:2.6.13} \end{equation}\]

Bu yerda so‘nish hadi Rayleigh turida deb olinadi va \(\eqref{eq:2.6.14}\) tenglamadagi kabi ifodalanadi.

\[\begin{equation} C = \alpha M + \beta K \label{eq:2.6.14} \end{equation}\]

Modal tahlildan olingan xos vektorlar yordamida t vaqtdagi ko‘chish vektori \(\eqref{eq:2.6.15}\) tenglamadagidek yoyilishi mumkin.

\[\begin{equation} U(t) = \sum_i b_i(t) U_i \label{eq:2.6.15} \end{equation}\]

Tashqi kuch hadi quyidagi garmonik shaklda bo‘lsin:

\[\begin{equation} F(t) = ( F_R + i F_I )e^{i \Omega t} \label{eq:2.6.16} \end{equation}\]

Endi \(b_{j}(t)\) ni aniqlaymiz. \(\eqref{eq:2.6.13}\) tenglamadagi harakat tenglamasi majburiy tebranish shakliga ega bo‘lgani uchun,

\[\begin{equation} b_j (t) = (b_{jR} + b_{jI}) e^{i \Omega t} \label{eq:2.6.17} \end{equation}\]

munosabat bajariladi. \(b_{j}(t)\) yoyilma koeffitsiyentining haqiqiy va mavhum qismlari mos ravishda \(\eqref{eq:2.6.18}i\) va \(\eqref{eq:2.6.19}\) tenglamalarda beriladi.

\[\begin{equation} b_{jR} = \frac{ U^T_j F_R (\omega^2_j - \Omega^2) + U^T_j F_I (\alpha + \beta \omega_j^2) \Omega}{ (\omega^2_j - \Omega^2)^2 + (\alpha + \beta \omega_j^2)^2 \Omega^2} \label{eq:2.6.18} \end{equation}\]
\[\begin{equation} b_{jI} = \frac{ U^T_j F_I(\omega^2_j - \Omega^2) - U^T_j F_R(\alpha + \beta \omega_j^2) \Omega}{ (\omega^2_j - \Omega^2)^2 + (\alpha + \beta \omega_j^2)^2 \Omega^2} \label{eq:2.6.19} \end{equation}\]

Shu bilan kerakli garmonik javob koeffitsiyentlari olinadi.

Tegishli mavzular