Lagrange's formulation can also be used for non$-$conservative systems by adding the applied non$-$conservative term to the right side of equation $(63)$ to get

$\frac{d}{dt}(\frac{\text{d}\text{e}\text{l}\text{T}}{del{q}_i^{}})-\frac{\text{d}\text{e}\text{l}\text{T}}{del{q}_i}+\frac{\text{d}\text{e}\text{l}\text{U}}{del{q}_i}+\frac{del{R}_i}{del{q}_i^{}}=0$

Here $R_i$ is the Rayleigh dissipation function defined in the case of a viscous damper attached to ground by

$R_i=\frac{1}{2}c{q}_i^{}{?}^2$

Use this extended Lagrange formulation to derive the equation of motion of the damped automobile suspension driven by a dynamometer illustrated in Figure P$79.$ Assume here that the dynamometer drives the system such that $x=r.$