so that ceqe=a1 is the optimal linear approximation. Equation (4.5) is often presented
without reference to the underlying orthonormal series, as in [32], where the method
is referred to as equivalent linearization. As an aside, it can be noted that if the
nonlinear damping is a function of the velocity, rather than an instantaneous,
memory-less function, then equation (4.5) would be replaced by a functional series
which is analogous to the Wiener series [33], and equation (4.7) would be replaced
by a cross-correlation approach analogous to the Lee–Schetzen algorithm [30].
The third and final approach to obtaining the quasi-linear damping coefficient is to
note that the power dissipation rates under linear and nonlinear damping are,
respectively,