where, q = β β sign ⁡ ⁡ ( z ( t ) u ˙ ˙ ( t ) ) + γ γ is assumed constant for the (not necessarily small) transition under examination, A = 1 and u 0 , z 0 are the initial values of the displacement and the hysteretic parameter, respectively. Eq.20 is solved analytically for z for specific values of the exponential parameter n , i.e. for n = 1 and n = 2 . For arbitrary values of n , Eq. 20 can be solved efficiently using e.g. bisection – type methods, such as the Brent's method.