Dyskretyzacja:

Zgodnie z tym przemieszczenia:

\begin{aligned} d=\left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ d_6 \\ 0 \\ 0 \\ d_9 \end{array}\right] \end{aligned}

Agregacja:

Jak łatwo zauważyć po wykreśleniu elementów macierz sztywności:

\begin{aligned} & \left[\begin{array}{ll} K_{66} & K_{69} \\ K_{96} & K_{99} \end{array}\right] = \left[\begin{array}{cc} k_{66.1} + k_{33.2} & k_{36.2} \\ k_{63.2} & k_{66.2} \end{array}\right] \\ \end{aligned} \begin{aligned} \begin{array}{ll} k_{66.1}=\frac{E \cdot I}{l_1{ }^3} \cdot 4 l_1{ }^2=6 \cdot 10^4 \\ k_{33.2}=\frac{E \cdot I}{l_2{ }^3} \cdot 4{l_2}^2=6 \cdot 10^4 & k_{36.2}=\frac{E \cdot I}{l_2{ }^3} \cdot 2 l_2{ }^2=3 \cdot 10^4 \\ k_{63.2}=\frac{E \cdot I}{l_2{ }^3} \cdot 2{l_2}^2=3 \cdot 10^4 & k_{66.2}=\frac{E \cdot I}{l_2{ }^3} \cdot 4 l_2{ }^2=6 \cdot 10^4 \end{array} \end{aligned}

Analogicznie macie mas:

\begin{aligned} & \left[\begin{array}{ll} M_{66} & M_{69} \\ M_{96} & M_{99} \end{array}\right] = \left[\begin{array}{cc} m_{66.1} + m_{33.2} & m_{36.2} \\ m_{63.2} & m_{66.2} \end{array}\right] \\ \end{aligned} \begin{aligned} \begin{array}{ll} m_{66.1}=\frac{\mu \cdot l_1}{420} \cdot 4 l_1^2=1.6 \\ m_{33.2}=\frac{\mu \cdot l_2}{420} \cdot 4 l_2^2=1.6 & m_{36.2}=\frac{\mu \cdot l_2}{420} \cdot\left(-3 l_2^2\right)=-1.2 \\ m_{63.2}=\frac{\mu \cdot l_2}{420} \cdot\left(-3 l_2^2\right)=-1.2 & m_{66.2}=\frac{\mu \cdot l_2}{420} \cdot 4 l_2^2=1.6 \end{array} \end{aligned}

Równanie dynamiki:

\begin{aligned} & {\left[\begin{array}{cc} k_{66.1}+k_{33.2} & k_{36.2} \\ k_{63.2} & k_{66.2} \end{array}\right]=\left[\begin{array}{rr} 120 & 30 \\ 30 & 60 \end{array}\right] 10^3} \\ & {\left[\begin{array}{cc} m_{66.1}+m_{33.2} & m_{36.2} \\ m_{63.2} & m_{66.2} \end{array}\right]=\left[\begin{array}{rr} 3.2 & -1.2 \\ -1.2 & 1.6 \end{array}\right]} \\ & {\left[\begin{array}{cc} 120 & 30 \\ 30 & 60 \end{array}\right] 10^3-\omega^2 \cdot\left[\begin{array}{cc} 3.2 & -1.2 \\ -1.2 & 1.6 \end{array}\right]=\left[\begin{array}{rr} -3.2 \cdot \omega^2+120000.0 & 1.2 \cdot \omega^2+30000.0 \\ 1.2 \cdot \omega^2+30000.0 & -1.6 \cdot \omega^2+60000.0 \end{array}\right]=0} \end{aligned}

Rozwiązania:

\begin{aligned} & \operatorname{det}\left(\left[\begin{array}{cc} -3.2 \cdot \omega^2+120000.0 & 1.2 \cdot \omega^2+30000.0 \\ 1.2 \cdot \omega^2+30000.0 & -1.6 \cdot \omega^2+60000.0 \end{array}\right]\right) \rightarrow 3.68 \cdot \omega^4-456000.0 \cdot \omega^2+6300000000.0 \\ & 3.68 \cdot \omega^4-456000.0 \cdot \omega^2+6300000000.0=0 \\ & 3.68 \cdot t^2-456000.0 \cdot t+6300000000.0=0 \\ & t_1=15.841 \cdot 10^3 \quad \omega_1=\sqrt{t_1}=125.861 \\ & t_2=108.072 \cdot 10^3 \quad \omega_2=\sqrt{t_2}=328.743 \end{aligned}

Obliczenia przemieszczeń:

\begin{aligned} & \left(K-\omega^2 M\right) d=0 \quad d_9=1 \\ & \left(\left[\begin{array}{cc} 120 & 30 \\ 30 & 60 \end{array}\right] 10^3-\omega_1{ }^2 \cdot\left[\begin{array}{cc} 3.2 & -1.2 \\ -1.2 & 1.6 \end{array}\right]\right) \cdot\left[\begin{array}{l} d_6 \\ d_9 \end{array}\right]=\left[\begin{array}{c} 49009.2 \cdot d_9+69308.8 \cdot d_6 \\ 34654.4 \cdot d_9+49009.2 \cdot d_6 \end{array}\right]=0 \\ & 49009.2 \cdot d_9+69308.8 \cdot d_6=0 \\ & d_6=-0.707 \\ & \left(K-\omega^2 M\right) d=0 \\ & \left(\left[\begin{array}{cc} 120 & 30 \\ 30 & 60 \end{array}\right] 10^3-\omega_2{ }^2 \cdot\left[\begin{array}{cc} 3.2 & -1.2 \\ -1.2 & 1.6 \end{array}\right]\right) \cdot\left[\begin{array}{l} d_6 \\ d_9 \end{array}\right]=\left[\begin{array}{c} 159686.0 \cdot d_9-225830.0 \cdot d_6 \\ -112915.0 \cdot d_9+159686.0 \cdot d_6 \end{array}\right]=0 \\ & -225830.0 \cdot d_6+159686.0=0 \\ & d_6=0.707 \end{aligned}

Szkicujemy przemieszczenia:

MES

Metody numeryczne - MES - Przykład 2

Przykład 2