If the constant ceqe is chosen to equate the ensemble average of the two
powers, E[PL]=E[PNL], then the result is
4.14
which is in agreement with equation (4.4). Thus, the mathematical optimization
technique leading to equation (4.14) ensures that a quantity of great physical
importance, the dissipated power, is conserved. In the expression for ceqe,
and can be interpreted as the cross-correlation between the
nonlinear damping force and the velocity and the auto-correlation function of the
squared velocity, both at zero lag, in agreement with Nuttall’s approach mentioned
above.
If the nonlinearity were to be in the stiffness, rather than the damping, then the
conserved quantity, in the equivalent to equation (4.14), would be E[h(x)x], which
does not have such physical significance, thus suggesting that the quasi-linear
approach can be expected to be more effective for nonlinear damping than for
nonlinear stiffness.