Dinamik tahlil usullari¶

Ushbu bo‘limda bevosita vaqt integratsiyasini qo‘llaydigan dinamik masalalarni tahlil qilish usullari ko‘rsatiladi. Quyida noaniq va oshkor sxemalarning formulalari keltiriladi.

Harakat tenglamasini diskretlashtirish (umumiy doira)¶

TBD (keyingi bosqichda matn yoziladi).

Noaniq sxema (Newmark-β usuli)¶

Dinamik masalalar uchun quyidagi harakat tenglamasini yechishda bevosita vaqt integratsiyasi usuli qo‘llanadi.

\[\begin{equation} M( t + \Delta t ) \ddot{U} (t + \Delta t) + C( t + \Delta t ) \dot{U}(t + \Delta t) + Q( t + \Delta t ) = F( t + \Delta t ) \label{eq:2.5.1} \end{equation}\]

Bu yerda \(M\) — massa matritsasi, \(C\) — so‘nish matritsasi, \(Q\) — ichki kuch vektori va \(F\) — tashqi kuch vektori. Massa matritsasi nochiziqli tahlilda ham deformatsiyaga bog‘liq bo‘lmay, doimiy deb olinadi.

Vaqt inkrementi \(\Delta t\) davomida siljish, tezlik va tezlanish o‘zgarishlari Newmark-\(\beta\) usuli bilan \(\eqref{eq:2.5.2}\) va \(\eqref{eq:2.5.3}\) tenglamalarda ko‘rsatilganidek yaqinlashtiriladi.

\[\begin{equation} \dot{U}(t + \Delta t) = \frac{\gamma}{\beta \Delta t} \Delta U( t + \Delta t ) - \frac{\gamma - \beta}{\beta} \dot{U}( t ) - \Delta t \frac{\gamma - 2\beta}{2\beta} \ddot{U}(t) \label{eq:2.5.2} \end{equation}\]

\[\begin{equation} \ddot{U}(t + \Delta t) = \frac{1}{\beta \Delta t^2}\Delta U(t + \Delta t) - \frac{1}{\beta \Delta t} \dot{U}(t) - \frac{1 - 2\beta}{2\beta} \ddot {U}(t) \label{eq:2.5.3} \end{equation}\]

Bu yerda \(\gamma\) va \(\beta\) Newmark-\(\beta\) usulining parametrlaridir.

Ma’lumki, \(\gamma\) va \(\beta\) quyidagi qiymatlarni olganda mos ravishda chiziqli tezlanish usuli yoki trapetsiya qoidasiga mos keladi.

\(\gamma = \displaystyle \frac{1}{2}\)，\(\beta = \displaystyle \frac{1}{6}\) (chiziqli tezlanish usuli)

\(\gamma = \displaystyle \frac{1}{2}\)，\(\beta = \displaystyle \frac{1}{4}\) (trapetsiya qoidasi)

\(\eqref{eq:2.5.2}\) va \(\eqref{eq:2.5.3}\) tenglamalarni \(\eqref{eq:2.5.1}\) ga qo‘yib, quyidagi tenglamani olamiz.

\[\begin{align} \nonumber \left( \frac{1}{\beta \Delta t^2} \mathbf{M} + \frac{\gamma}{\beta \Delta t} C + K \right) \Delta U ( t + \Delta t ) &= F ( t + \Delta t ) - Q ( t + \Delta t ) \\\ \nonumber &+ \frac{1}{\beta \Delta t} M \dot{U} ( t ) + \frac{1 - 2\beta}{2\beta} M \ddot{U} ( t ) \\\ &+ \frac{\gamma - \beta}{\beta} C \dot{U} (t) + \Delta t \frac{\gamma - 2\beta}{2 \beta} C \ddot{U}(t) \label{eq:2.5.4} \end{align}\]

Xususan, chiziqli masala uchun \(K_L\) chiziqli qattiqlik matritsasi bo‘lib, \(Q ( t + \Delta t ) = K_L U (t + \Delta t)\). Ushbu munosabatni yuqoridagi tenglamaga qo‘ysak, quyidagi tenglama olinadi.

\[\begin{align} \nonumber M \left\lbrace -\frac{1}{\beta \Delta t^2} U(t) -\frac{1}{\beta \Delta t}\dot U(t) - \frac{2\beta}{1-2\beta} \ddot U(t) \right\rbrace &+ C\left\lbrace - \frac{\gamma}{\beta \Delta t} U(t) + \left(1 - \frac{\gamma}{\beta}\right) \dot U(t) + \Delta{t}\frac{ 2\beta-\gamma}{2\beta}\ddot U(t)\right\rbrace \\\ & + \frac{1}{\beta \Delta{t}^2} M + \frac{\gamma}{\beta \Delta{t}} C + K_L U(t+\Delta{t}) = F(t+\Delta{t}) \label{eq:2.5.5} \end{align}\]

Geometrik chegaraviy shart sifatida tezlanish belgilangan joylarda \(\eqref{eq:2.5.2}\) tenglamadan quyidagi siljish olinadi.

\[\begin{equation} u_{is} (t+\Delta{t}) = u_{is} (t) + \Delta t \dot{u}(t) + \Delta t^2 \left(\frac{1}{2} -\beta \right) {\ddot{u}}_{is} (t + \Delta t) \label{eq:2.5.6} \end{equation}\]

Xuddi shuningdek, tezlik belgilangan joylarda \(\eqref{eq:2.5.6}\) tenglamadan quyidagi siljish olinadi.

\[\begin{equation} u_{is}(t+\Delta{t})= u_{is}(t)+\Delta t \frac{ \gamma - \beta}{ \gamma}\dot{u_{is}}(t) +(\Delta{t})^2 \frac{ \gamma - 2\beta}{ 2\gamma} \ddot{u_{is}}(t) +\Delta t \frac{\beta}{ \gamma}\dot{u_{is}}(t+\Delta{t}) \label{eq:2.5.7} \end{equation}\]

Bu yerda, \(u_{is}(t+\Delta{t})\) — \(t+\Delta{t}\) vaqtdagi tugun siljishi, \(\dot{u_{is}}(t+\Delta{t})\) — \(t+\Delta{t}\) vaqtdagi tugun tezligi, \(\ddot{u_{is}}(t+\Delta{t})\) — \(t+\Delta{t}\) vaqtdagi tugun tezlanishi, \(i\) — tugunning erkinlik darajasi raqami, \(s\) — tugun raqami. Massa va so‘nish hadlari quyidagicha ko‘rib chiqiladi.

Massa hadini ko‘rib chiqish¶

Massa matritsasi, qoida tariqasida, jamlangan massa matritsasi sifatida olinadi.

So‘nish hadini ko‘rib chiqish¶

So‘nish hadi \(\eqref{eq:2.5.8}\) tenglama bilan ifodalangan Rayleigh so‘nishi sifatida olinadi.

\[\begin{equation} C = R_m M + R_k K_L \label{eq:2.5.8} \end{equation}\]

Bu yerda \(R_m\) va \(R_k\) Rayleigh so‘nishining parametrlaridir.

!DYNAMIC kartasida ko‘rsatilgan \(R_m\) va \(R_k\) butun modelga bir xil qo‘llanadi. Har bir material uchun turlicha \(R_m\) va \(R_k\) berish kerak bo‘lsa, shu materialning !MATERIAL blokida !DAMPING kartasini ko‘rsating. !DAMPING ko‘rsatilgan materialga tegishli elementlar uchun element so‘nish matritsasi element massa matritsasi \(M_i\) va tangensial qattiqlik matritsasi \(K_i\) dan \(C_i = R_m M_i + R_k K_i\) sifatida hisoblanib, global so‘nish matritsasiga yig‘iladi. Bu funksiya faqat noaniq sxemada amal qiladi.

Oshkor sxema (markaziy ayirmalar usuli)¶

Oshkor sxema quyida ko‘rsatilgan t vaqtdagi harakat tenglamasiga asoslanadi.

\[\begin{equation} M \ddot{U}(t) + C (t) \dot{U}(t) + Q(t) = F(t) \label{eq:2.5.9} \end{equation}\]

Bu yerda \(t + \Delta t\) va \(t - \Delta t\) vaqtlardagi siljishlar \(t\) vaqtdagi Taylor qatoriga yoyiladi va \(\Delta t\) bo‘yicha ikkinchi tartibli hadlargacha olinsa, quyidagilar hosil bo‘ladi.

\[\begin{equation} U(t+\Delta{t}) = U(t)+\dot{U}(t)(\Delta{t}) +\frac{1}{2!}\ddot{U}(\Delta{t})^2 \label{eq:2.5.10} \end{equation}\]

\[\begin{equation} U(t-\Delta{t})=U(t)-\dot{U}(t)(\Delta{t}) +\frac{1}{2!}\ddot{U}(\Delta{t})^2 \label{eq:2.5.11} \end{equation}\]

\(\eqref{eq:2.5.3}\) va \(\eqref{eq:2.5.4}\) tenglamalarning ayirmasi va yig‘indisidan quyidagi tenglamalar olinadi.

\[\begin{equation} \dot{U}(t)=\frac{1}{2\Delta{t}} (U(t+\Delta{t})-U(t-\Delta{t})) \label{eq:2.5.12} \end{equation}\]

\[\begin{equation} \ddot{U}= \frac{1}{(2\Delta{t})^2} (U(t+\Delta{t})-2U(t)+U(t-\Delta{t})) \label{eq:2.5.13} \end{equation}\]

\(\eqref{eq:2.5.12}\) va \(\eqref{eq:2.5.13}\) tenglamalarni \(\eqref{eq:2.5.9}\) ga qo‘yib, quyidagi tenglamani olamiz.

\[\begin{equation} \left( \frac{1}{\Delta t^{2}} M + \frac{1}{2\Delta t} C \right) U ( t + \Delta t ) \\\ = F(t) - Q(t) - \frac{1}{\Delta t^{2}} 2 U(t) - U( t - \Delta t) - \frac{1}{2\Delta t} C U(t - \Delta t) \label{eq:2.5.14} \end{equation}\]

Xususan, chiziqli masala uchun \(Q(t) = K_L U(t)\) bo‘ladi va yuqoridagi tenglama quyidagi ko‘rinishga keladi.

\[\begin{equation} \left( \frac{1}{\Delta t^{2}} M + \frac{1}{2\Delta t} C \right) U( t + \Delta t ) \\\ = F(t) - K_L U(t) - \frac{1}{\Delta t^{2}} M U(t) - U(t - \Delta t) - \frac{1}{2\Delta t} C U (t - \Delta t) \label{eq:2.5.15} \end{equation}\]

Bu yerda massa matritsasi \(M\) jamlangan massa matritsasi, so‘nish matritsasi esa \(C = R_m M\) proporsional so‘nish matritsasi deb olinsa, \(\eqref{eq:2.5.15}\) tenglamada tenglamalar sistemasini yechish talab qilinmaydi.

Shuning uchun \(\eqref{eq:2.5.15}\) tenglamadan \(U(t+\Delta t)\) ni quyidagi tenglama bilan topish mumkin.

\[\begin{equation} U( t + \Delta t ) \\\ = \frac{1}{( \frac{1}{\Delta t^{2}} M + \frac{1}{2\Delta t} C )} \{ F(t) - Q(t) - \frac{1}{\Delta t^{2}} M U(t) - U(t - \Delta t) - \frac{1}{2\Delta t} C U(t - \Delta t) \} \label{eq:2.5.17} \end{equation}\]

Tegishli mavzular¶

Tahlil turlari — dinamik tahlil funksiyalarining umumiy ko‘rinishi va ularni tanlash
Adabiyotlar