The characteristic roots of the model jump from Ao| ol? = 1.2 | to 
A al ols 0.96 or vice versa as in Fig. 13.6 and also shown by a step function in 
Fig. 13.8. While in the model as is Eq. 13.53, the path of two rootsAp andA and 4 at 
arbitrary X,_; is specified by a continuous Hermite—type polynomials in the equation, as is 
shown by a continuous curve in Fig. 13.8. 
Al? 
05 1.0 Ka 
Fig. 13.8. Path of characteristic roots of threshold model and 
exponential AR model. 
(From Ozaki.99) 
Ozaki argued that the threshold linear AR model, however, could not be expected to 
give a good enough approximation to nonlinear vibration, and appropriate nonlinear 
threshold AR model must be formulated. His point is, we have to consider the stepwise 
dynamics of restoring and damping force, but at the same time, the orbitally stableness 
and independence of the limit cycle of the Van del Pol equation must be maintained. 
Instead of the linear step function approximation, nonlinear approximation 
considered was as follows. 
r= 
{ 1.95 X,_1 — 0.96 Xp2+ Er, for IX ,-1! => 1.0 
(2.18 =0.23 X2,) Xe1—(1.2—0.24 X2) X,o4€:, 
tae (Osh 2 10) (13.58) 
Equation 13.58 can give the characteristic roots that move much more smoothly than 
those of Eq. 13.57. Thus, the general form of the nonlinear threshold autoregressive mod- 
el for nonlinear vibrations was proposed as, 
ae (ae ° “+@p Xipt€:, for \X,_4| =>T (13.59) 
where 
fA) = GP + aPX4-- +0 XM. 
i 
Schematically, the behavior of the characteristic roots for (1) linear threshold model, 
(2) nonlinear threshold model, and (3) exponential AR model are expressed as shown in 
Fig. 13.9. 
390 
