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## Q. 2.10

The system of Figure 2.24 moves in a horizontal plane.
(a) Determine the equivalent viscous-damping coefficient for the system if x is the displacement of the 2 kg block and is used as the generalized coordinate.
(b) Determine the equivalent, torsional viscous-damping coefficient $\theta$ if the clockwise angular displacement of the disk is used as the generalized coordinate. ## Verified Solution

(a) Using kinematics, it is found that the relation between the downward displacement of the 2 kg block x and the upward displacement of the 1 kg block y is $y= \frac{3}{2} x$. Calculating the work done by the viscous dampers as the system moves between the initial position and an arbitrary position, we have

$U_{1\rightarrow 2}=-\int_{0}^{x}{(200 N\cdot s/m)}\dot{x}dx – \int_{0}^{x}{(400 N\cdot s/m)}\left(\frac{3}{2}\dot{x}\right)d\left(\frac{3}{2}x\right)$

$=-\int_{0}^{x}{(1100 N\cdot s/m)}\dot{x}dx$                     (a)

Thus, $c_{eq}=1100 N\cdot s/n$

b) Kinematics is used to determine that $x=r\theta$ and $y=\frac{3}{2}r\theta$ where r=0. Calculating the work done by the viscous dampers as the system moves from an initial position to an arbitrary position, we have

$U_{1\rightarrow 2}=-\int_{0}^{\theta }{(200 N\cdot s/m)}\left[\left(0.1m\right) \dot{\theta } \right] d\left[\left(0.1m\right) \theta \right]- \int_{0}^{\theta }{(400 N\cdot s/m)}\left[\frac{3}{2} \left(0.1m\right) \dot{\theta } \right]$

$\times d\left[\frac{3}{2} \left(0.1m\right) \theta \right]=-\int_{0}^{\theta }{(11\frac{N\cdot m\cdot s}{rad} )\dot{\theta }d\theta }$           (b)

Thus, $c_{t,eq}=11 N\cdot m\cdot s/rad$