The damped free vibration of a two degree of freedom system moving in the xy -plane has total kinetic energy [latex]T=\frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{y}^2[/latex], total elastic potential energy [latex]V=\frac{1}{2} k_1 x^2 + \frac{1}{2} k_2 y^2[/latex], and a total Stokes type dissipation described by the Rayleigh function [latex]D=\frac{1}{2}\left(c_1 \dot{x}^2 + 2 c_{12} \dot{x} \dot{y} + c_2 \dot{y}^2\right) .[/latex] Derive the equations of motion.

The Lagrangian is [latex]L=\frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{y}^2 - \frac{1}{2} k_1 x^2 - \frac{1}{2} k_2 y^2[/latex]. For the free vibrational motion [latex]Q_r^N=0[/latex] and use of L and D in (11.99) yields the two equations of motion for the system: [latex]\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_r}\right)-\frac{\partial L}{\partial q_r}+\frac{\partial D}{\partial \dot{q}_r}=Q_r^N \text {. }[/latex] (11.99) [latex]\begin{aligned}&m_1 \ddot{x} + c_1 \dot{x} + c_{12} \dot{y} + k_1 x=0, \\&m_2 \ddot{y} + c_{12} \dot{x} + c_2 \dot{y} + k_2 y=0\end{aligned}.[/latex] (11.100) This is a coupled system of linear differential equations for which general solution methods are well known. See Whittaker in the References for further study of this topic .

Question 11.14: The damped free vibration of a two degree of freedom system ...

Principles of Engineering Mechanics: Volume 2 Dynamics – The Analysis of Motion

The damped free vibration of a two degree of freedom system moving in the xy -plane has total kinetic energy $T=\frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{y}^2$ , total elastic potential energy $V=\frac{1}{2} k_1 x^2 + \frac{1}{2} k_2 y^2$ , and a total Stokes type dissipation described by the Rayleigh function $D=\frac{1}{2}\left(c_1 \dot{x}^2 + 2 c_{12} \dot{x} \dot{y} + c_2 \dot{y}^2\right) .$ Derive the equations of motion.

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The Lagrangian is $L=\frac{1}{2} m_1 \dot{x}^2 + \frac{1}{2} m_2 \dot{y}^2 – \frac{1}{2} k_1 x^2 – \frac{1}{2} k_2 y^2$ . For the free vibrational motion $Q_r^N=0$ and use of L and D in (11.99) yields the two equations of motion for the system:

$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_r}\right)-\frac{\partial L}{\partial q_r}+\frac{\partial D}{\partial \dot{q}_r}=Q_r^N \text {. }$ (11.99)

$\begin{aligned}&m_1 \ddot{x} + c_1 \dot{x} + c_{12} \dot{y} + k_1 x=0, \\&m_2 \ddot{y} + c_{12} \dot{x} + c_2 \dot{y} + k_2 y=0\end{aligned}.$ (11.100)

This is a coupled system of linear differential equations for which general solution methods are well known. See Whittaker in the References for further study of this topic .