Dyskretyzacja:

Zgodnie z tym przemieszczenia:

\begin{aligned} d=\left[\begin{array}{c} 0 \\ 0 \\ d_3 \\ 0 \\ 0 \\ d_6 \\ 0 \\ 0 \\ 0 \end{array}\right] \end{aligned}

Agregacja:

Jak łatwo zauważyć po wykreśleniu elementów macierz sztywności:

\begin{aligned} \begin{bmatrix} K_{33} & K_{36} \\ K_{63} & K_{66} \end{bmatrix} = \begin{bmatrix} k_{33.1} & k_{36.1} \\ k_{63.1} & k_{66.1} + k_{33.2} \end{bmatrix} \end{aligned} \begin{aligned} k_{33.1} &= \frac{E \cdot I}{l_1^3} \cdot 4 \cdot l_1^2 = 2 \cdot 10^4 \end{aligned} \begin{aligned} k_{36.1} &= \frac{E \cdot I}{l_1^3} \cdot 2 \cdot l_1^2 = 1 \cdot 10^4 \end{aligned} \begin{aligned} k_{63.1} &= \frac{E \cdot I}{l_1^3} \cdot 2 \cdot l_1^2 = 1 \cdot 10^4 \end{aligned} \begin{aligned} k_{66.1} &= \frac{E \cdot I}{l_1^3} \cdot 4 \cdot l_1^2 = 2 \cdot 10^4 \end{aligned} \begin{aligned} k_{33.2} &= \frac{E \cdot I}{l_2^3} \cdot 4 \cdot l_2^2 = 3 \cdot 10^4 \end{aligned}

Analogicznie macierz mas:

\begin{aligned} \begin{bmatrix} M_{33} & M_{36} \\ M_{63} & M_{66} \end{bmatrix} = \begin{bmatrix} m_{33.1} & m_{36.1} \\ m_{63.1} & m_{66.1} + m_{33.2} \end{bmatrix} \end{aligned} \begin{aligned} m_{33.1} &= \frac{\mu \cdot l_1}{420} \cdot 4 \cdot l_1^2 = 10.8 \end{aligned} \begin{aligned} m_{36.1} &= \frac{\mu \cdot l_1}{420} \cdot (-3 \cdot l_1^2) = -8.1 \end{aligned} \begin{aligned} m_{63.1} &= \frac{\mu \cdot l_1}{420} \cdot (-3 \cdot l_1^2) = -8.1 \end{aligned} \begin{aligned} m_{66.1} &= \frac{\mu \cdot l_1}{420} \cdot 4 \cdot l_1^2 = 10.8 \end{aligned} \begin{aligned} m_{33.2} &= \frac{\mu \cdot l_2}{420} \cdot 4 \cdot l_2^2 = 3.2 \end{aligned}

Równanie dynamiki:

\begin{aligned} \begin{bmatrix} k_{33.1} & k_{36.1} \\ k_{63.1} & k_{66.1} + k_{33.2} \end{bmatrix} - \omega^2 \cdot \begin{bmatrix} m_{33.1} & m_{36.1} \\ m_{63.1} & m_{66.1} + m_{33.2} \end{bmatrix} = \begin{bmatrix} -10.8 \cdot \omega^2 + 20000.0 & 8.1 \cdot \omega^2 + 10000.0 \\ 8.1 \cdot \omega^2 + 10000.0 & -14.0 \cdot \omega^2 + 50000.0 \end{bmatrix} \end{aligned} \begin{aligned} \text{det}\left( \begin{bmatrix} -10.8 \cdot \omega^2 + 20000.0 & 8.1 \cdot \omega^2 + 10000.0 \\ 8.1 \cdot \omega^2 + 10000.0 & -14.0 \cdot \omega^2 + 50000.0 \end{bmatrix} \right) = 0 \end{aligned}

Rozwiązania:

\begin{aligned} 85.59 \cdot \omega^4 - 982000.0 \cdot \omega^2 + 900000000.0 &= 0 \end{aligned} \begin{aligned} 85.59 \cdot t^2 - 982000.0 \cdot t + 900000000.0 &= 0 \end{aligned} \begin{aligned} t_1 &= 1004.43 \quad \omega_1 = \sqrt{t_1} = 31.693 \\ t_2 &= 10468.873 \quad \omega_2 = \sqrt{t_2} = 102.318 \end{aligned}

Obliczenia przemieszczeń:

\begin{aligned} (K - \omega^2 M) \cdot d &= 0 \quad d_3 = 1 \end{aligned} \begin{aligned} \left( \begin{bmatrix} k_{33.1} & k_{36.1} \\ k_{63.1} & k_{66.1} + k_{33.2} \end{bmatrix} - \omega_1^2 \cdot \begin{bmatrix} m_{33.1} & m_{36.1} \\ m_{63.1} & m_{66.1} + m_{33.2} \end{bmatrix} \right) \cdot \begin{bmatrix} d_3 \\ d_6 \end{bmatrix} &= \begin{bmatrix} 18136.0 \cdot d_6 + 9152.2 \cdot d_3 \\ 35938.0 \cdot d_6 + 18136.0 \cdot d_3 \end{bmatrix} = 0 \end{aligned} \begin{aligned} 18136.0 \cdot d_6 + 9152.2 &= 0 \\ d_6 &= -0.505 \end{aligned} \begin{aligned} (K - \omega^2 M) \cdot d &= 0 \quad d_3 = 1 \end{aligned} \begin{aligned} \left( \begin{bmatrix} k_{33.1} & k_{36.1} \\ k_{63.1} & k_{66.1} + k_{33.2} \end{bmatrix} - \omega_2^2 \cdot \begin{bmatrix} m_{33.1} & m_{36.1} \\ m_{63.1} & m_{66.1} + m_{33.2} \end{bmatrix} \right) \cdot \begin{bmatrix} d_3 \\ d_6 \end{bmatrix} &= \begin{bmatrix} 94800.0 \cdot d_6 - 93060.0 \cdot d_3 \\ -96560.0 \cdot d_6 + 94800.0 \cdot d_3 \end{bmatrix} = 0 \end{aligned} \begin{aligned} 94800.0 \cdot d_6 - 93060.0 &= 0 \\ d_6 &= 0.982 \end{aligned}

Szkicujemy przemieszczenia:

MES

Metody numeryczne - MES - Przykład 8

Przykład 8