Global and Local Nonlinear System Responses under Narrowband Random Excitations. I: Semianalytical Method Solomon C. Yim, M.ASCE 1 ; Dongjun Yuk 2 ; Arvid Naess, M.ASCE 3 ; and I-Ming Shih 4 Abstract: A single-degree-of-freedom nonlinear structural system under narrowband random excitation can exhibit very complex global and local response behaviors. In order to develop a stochastic method to analyze the nonlinear responses, the system under deterministic excitation is first modeled and examined in the primary and subharmonic resonance regions. Typical response behaviors including coexistence of attractors and global jump phenomena are observed. Governing equations of the probability for the response–amplitude perturbations a local transition within an attraction domain and a jump between different attraction domains a global transition are derived under the assumption of a stationary Markov condition. The overall response–amplitude probability distribution is obtained by applying the Bayes formula to the two types of response transition probability distributions. In this study, we focus on understanding the physics of the transitions using the proposed probability method. DOI: 10.1061/ASCE0733-93992007133:122 CE Database subject headings: Nonlinear response; Excitation; Oscillations; Structural behavior; Probabilistic methods. Introduction Dynamic system response behaviors under narrowband stochastic excitations have been studied for decades Rice 1954; Lyon et al. 1961; Dimentberg 1971; Richard and Anand 1983; Davies and Liu 1990; Koliopulos and Bishop 1993. To date, the stochastic behavior of linear systems is well understood by using the spec- tral analysis technique Crandall and Mark 1963; Lin 1967; Nigam 1983; Roberts and Spanos 1990; Newland 1993; Soong and Grigoriu 1993; Lutes and Sarkani 1997. However, relatively little understanding on the stochastic behavior of nonlinear sys- tems subjected to narrowband excitations has been achieved be- cause of the complexity of the system response characteristics. The complex response behavior of a nonlinear mechanical or structural system under deterministic excitations includes the jump phenomenon a global bifurcation behavior, sub- harmonic and superharmonic resonance, and even chaotic re- sponse coexisting attraction domainsNayfeh and Mook 1979; Guckenheimer and Holmes 1986; Thompson and Stewart 1986; Jordan and Smith 1999. To investigate these complex responses, semianalytical methods and numerical techniques Gottlieb and Yim 1992; Yim and Lin 1992 are required in general. In recent years, Roberts and Spanos 1986 and Davies and Liu 1990 approximated the excitation and response as Markov processes and pointed out a general rule of applying the stochastic averaging method in analyzing the stochastic system response under narrowband excitations. By solving the associated Fokker– Planck equation relating the excitation envelope and the response envelope, an approximate probability density function of the response envelope process was obtained. A direct numerical solu- tion of the Fokker–Planck equation using the path integral proce- dure was developed by Naess and Johnsen 1993see also Naess 1994, 1997. The method was successfully applied to the analysis and predictions of nonlinear system responses by Naess and Moe 2000 and Yim et al. 2005a and b. Probabilities of exceedence were calculated by integrating the four-dimensional state space process formed by the response displacement and velocity and the second order filter approximating the force excitation spectrum. Although the path integral solution method provides accurate nu- merical data in the form of joint probability densities, the results have to be presented in graphical form and are not convenient for understanding the physics of the response behaviors. Alternatively, a quasi-harmonic method was introduced by Koliopulos and Bishop 1993. Under the assumption that both the excitation and response processes are narrowband, the quasi- harmonic analysis leads to the formulation of a probability den- sity function of the response envelope. However, the excitation bandwidth effect on the response behavior is not taken into ac- count by this method, although the validity of this method can be determined by an extra parameter that indicates the occurrence and persistence of the response amplitude jump phenomenon. In previous studies, the analytical methods developed were based on deterministic techniques. However, due to the random nature of stochastic excitation processes e.g., waves in the ocean, the system response should be considered as in a transient state, and nearly steady-state behavior should be a special case 1 Professor, Dept. of Civil Engineering, Oregon State Univ., Corvallis, OR 97331-2302 corresponding author. E-mail: solomon.yim@orst.edu 2 Post-Doctoral Fellow, Dept. of Civil Engineering, Oregon State Univ., Corvallis, OR 97331-2302. 3 Professor, Dept. of Mathematical Science, Norwegian Univ. of Science and Technology, NO-7491 Trondheim, Norway. 4 Dept. of Civil Engineering, Oregon State Univ., Corvallis, OR 97331-2302; formerly, Graduate Research Assistant. Note. Associate Editor: Ross Barry Corotis. Discussion open until June 1, 2007. Separate discussions must be submitted for individual pa- pers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on December 19, 2005; approved on April 19, 2006. This paper is part of the Journal of Engi- neering Mechanics, Vol. 133, No. 1, January 1, 2007. ©ASCE, ISSN 0733-9399/2007/1-22–29/$25.00. 22 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JANUARY 2007