Dinamiese analise Metodes¶

Hierdie afdeling beskryf analise metodes vir dinamies probleme met behulp van direk tydintegrasie. Die formulerings van die implisiet en eksplisiet metodes is aangebied hieronder.

Diskretisering van die Bewegingsvergelyking (Algemeen Raamwerk)¶

TBD (na wees voltooi in die volgende phase).

Implisiet Metode (Newmark-β Metode)¶

Vir dinamies probleme, 'n direk tydintegrasie metode is toegepas na solve die bewegingsvergelyking hieronder getoon.

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

Hier, \(M\) is die massamatriks, \(C\) is die demping matriks, \(Q\) is die interne-kragvektor, en \(F\) is die eksterne-kragvektor. Die massamatriks is aangeneem na bly konstant ongeag vervorming, selfs in nie-lineêr analise.

Die veranderings in verplasing, snelheid, en versnelling oor die tydinkrement \(\Delta t\) is approximated met behulp van die Newmark-\(\beta\) metode as getoon in Eq. \(\eqref{eq:2.5.2}\) en Eq. \(\eqref{eq:2.5.3}\).

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

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

Hier, \(\gamma\) en \(\beta\) is parameters van die Newmark-\(\beta\) metode.

As is goed bekend, die volgende waardes van \(\gamma\) en \(\beta\) stem ooreen na die lineêr versnelling metode en die trapezoidal reël, onderskeidelik.

\(\gamma = \displaystyle \frac{1}{2}\), \(\beta = \displaystyle \frac{1}{6}\) (lineêr versnelling metode)

\(\gamma = \displaystyle \frac{1}{2}\), \(\beta = \displaystyle \frac{1}{4}\) (trapezoidal reël)

Deur in te vul Eq. \(\eqref{eq:2.5.2}\) en Eq. \(\eqref{eq:2.5.3}\) in Eq. \(\eqref{eq:2.5.1}\) gee die volgende vergelyking.

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

In spesifieke, vir 'n lineêr probleem, \(K_L\) is die lineêr styfheidsmatriks en \(Q ( t + \Delta t ) = K_L U (t + \Delta t)\). Deur in te vul hierdie verhouding in die vergelyking hierbo gee die volgende vergelyking.

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

By liggings waar versnelling is voorgeskrewe as 'n geometries randvoorwaarde, die verplasing word verkry van Eq. \(\eqref{eq:2.5.2}\) soos volg.

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

Similarly, by liggings waar snelheid is voorgeskrewe, die verplasing word verkry van Eq. \(\eqref{eq:2.5.6}\) soos volg.

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

Hier, \(u_{is}(t+\Delta{t})\) is die knoop- verplasing by tyd \(t+\Delta{t}\), en \(\dot{u_{is}}(t+\Delta{t})\) is die knoop- snelheid by tyd \(t+\Delta{t}\), \(\ddot{u_{is}}(t+\Delta{t})\) is die knoop- versnelling by tyd \(t+\Delta{t}\), \(i\) is die knoop- vryheidsgraad aantal, en \(s\) is die knoopnommer. Die massa en demping terme is hanteer soos volg.

Hantering van die Massa Term¶

In beginsel, die massamatriks is behandel as 'n lumped massamatriks.

Hantering van die Demping Term¶

Die demping term is behandel as Rayleigh demping uitgedruk deur Eq. \(\eqref{eq:2.5.8}\).

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

Hier, \(R_m\) en \(R_k\) is die Rayleigh demping parameters.

Die \(R_m\) en \(R_k\) waardes gespesifiseer op die !DYNAMIC kaart is toegepas uniformly na die hele model. Na assign verskillende \(R_m\) en \(R_k\) waardes vir elke materiaal, spesifiseer die !MATERIAL kaart binne dat materiaal's !DAMPING blok. Vir elemente behorende na 'n materiaal vir wat !DAMPING word gespesifiseer, die element demping matriks word bereken as \(M_i\) van die element massamatriks \(K_i\) en raaklynstyfheidsmatriks \(C_i = R_m M_i + R_k K_i\), en saamgestel in die globaal demping matriks. Hierdie funksie is effective slegs vir die implisiet metode.

Eksplisiet Metode (Central Verskil Metode)¶

Die eksplisiet metode is gebaseer op die bewegingsvergelyking by tyd t hieronder getoon.

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

Expressing die verplasings by kere \(t + \Delta t\) en \(t - \Delta t\) deur Taylor expansions oor tyd \(t\) en retaining terme deur tweede orde in \(\Delta t\) gee die volgende vergelykings.

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

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

Taking die verskil en sum van Eq. \(\eqref{eq:2.5.3}\) en Eq. \(\eqref{eq:2.5.4}\) gee die volgende vergelykings.

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

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

Deur in te vul Eq. \(\eqref{eq:2.5.12}\) en Eq. \(\eqref{eq:2.5.13}\) in Eq. \(\eqref{eq:2.5.9}\) gee die volgende vergelyking.

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

In spesifieke, vir 'n lineêr probleem, \(Q(t) = K_L U(t)\), en die vergelyking hierbo word

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

Indien die massamatriks \(M\) is taken as 'n lumped massamatriks en die demping matriks as 'n eweredig demping matriks \(C = R_m M\), Eq. \(\eqref{eq:2.5.15}\) vereis geen oplossing van simultaneous vergelykings.

Daarom, van Eq. \(\eqref{eq:2.5.15}\), \(U(t+\Delta t)\) kan wees verkry deur die volgende vergelyking.

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

Verwante onderwerpe¶

Analisetipes — Oorsig en keuse van dinamiese-analise funksies
Verwysings