Dyskretyzacja:

Zgodnie z tym przemieszczenia:

\begin{aligned} d=\left[\begin{array}{c} 0 \\ d_2 \\ 0 \\ d_4 \\ 0 \\ 0 \\ 0 \\ 0 \\ d_9 \end{array}\right] \end{aligned}

Agregacja:

Jak łatwo zauważyć po wykreśleniu elementów macierz sztywności:

\begin{aligned} & \left[\begin{array}{lll} K_{22} & K_{24} & K_{29} \\ K_{42} & K_{44} & K_{49} \\ K_{49} & K_{94} & K_{99} \end{array}\right] = \left[\begin{array}{ccc} k_{22.1} & k_{24.1} & 0 \\ k_{42.1} & k_{44.1} + k_{11.2} & k_{16.2} \\ 0 & k_{61.2} & k_{66.2} \end{array}\right] \\ & k_{22.1} = \frac{E \cdot I}{l_1^3} \cdot 12 = 2.4 \cdot 10^4 \quad k_{11.2} = \frac{E \cdot I}{l_2^3} \cdot \frac{A \cdot l_2^2}{I} = 3 \cdot 10^5 \\ & k_{24.1} = 0 \quad k_{16.2} = 0 \\ & k_{42.1} = 0 \quad k_{61.2} = 0 \\ & k_{44.1} = \frac{E \cdot I}{l_1^3} \cdot \frac{A \cdot l_1^2}{I} = 2 \cdot 10^5 \quad k_{66.2} = \frac{E \cdot I}{l_2^3} \cdot 4 l_2^2 = 1.08 \cdot 10^5 \end{aligned}

Analogicznie macie mas:

\begin{aligned} & \left[\begin{array}{lll} M_{22} & M_{24} & M_{29} \\ M_{42} & M_{44} & M_{49} \\ M_{49} & M_{94} & M_{99} \end{array}\right] = \left[\begin{array}{ccc} m_{22.1} & m_{24.1} & 0 \\ m_{42.1} & m_{44.1} + m_{11.2} & m_{16.2} \\ 0 & m_{61.2} & m_{66.2} \end{array}\right] \\ & m_{22.1} = \frac{\mu \cdot l_1}{420} \cdot 156 = 23.4 \quad m_{11.2} = \frac{\mu \cdot l_2}{420} \cdot 140 = 14 \\ & m_{24.1} = 0 \quad m_{16.2} = 0 \\ & m_{42.1} = 0 \quad m_{61.2} = 0 \\ & m_{44.1} = \frac{\mu \cdot l_1}{420} \cdot 140 = 21 \quad m_{66.2} = \frac{\mu \cdot l_2}{420} \cdot 4 l_2^2 = 1.6 \end{aligned}

Równanie dynamiki:

\begin{aligned} & \left[\begin{array}{ccc} k_{22.1} & k_{24.1} & 0 \\ k_{42.1} & k_{44.1} + k_{11.2} & k_{16.2} \\ 0 & k_{61.2} & k_{66.2} \end{array}\right] = \left[\begin{array}{rrr} 24 & 0 & 0 \\ 0 & 500 & 0 \\ 0 & 0 & 108 \end{array}\right] \cdot 10^3 \\ & \left[\begin{array}{ccc} m_{22.1} & m_{24.1} & 0 \\ m_{42.1} & m_{44.1} + m_{11.2} & m_{16.2} \\ 0 & m_{61.2} & m_{66.2} \end{array}\right] = \left[\begin{array}{ccc} 23.4 & 0 & 0 \\ 0 & 35 & 0 \\ 0 & 0 & 1.6 \end{array}\right] \\ & \left[\begin{array}{ccc} 24 & 0 & 0 \\ 0 & 500 & 0 \\ 0 & 0 & 108 \end{array}\right] \cdot 10^3 - \omega^2 \cdot \left[\begin{array}{ccc} 23.4 & 0 & 0 \\ 0 & 35 & 0 \\ 0 & 0 & 1.6 \end{array}\right] = \left[\begin{array}{ccc} -23.4 \cdot \omega^2 + 24000 & 0 & 0 \\ 0 & -35 \cdot \omega^2 + 500000 & 0 \\ 0 & 0 & -1.6 \cdot \omega^2 + 108000 \end{array}\right] = 0 \end{aligned}

Rozwiązania:

\begin{aligned} \begin{array}{ll} -23.4 \cdot \omega^2 + 24000 = 0 & \omega_1 = \sqrt{\frac{24000}{23.4}} = 32.026 \\ -35 \cdot \omega^2 + 500000 = 0 & \omega_2 = \sqrt{\frac{500000}{35}} = 119.523 \\ -(-1.6 \cdot \omega^2 + 108000) = 0 & \omega_3 = \sqrt{\frac{108000}{1.6}} = 259.808 \end{array} \end{aligned}