## 1 Introduction  
Generally, experimental modal analysis was performed to ensure structural stability under operating conditions. The structure's stability under dynamic conditions was evaluated based upon the identified modal parameters, such as the natural frequency, mode shape, and damping ratio. However, most mechanical structures do not operate under steady-state conditions; moreover, the operating frequency varies with the magnitude of the excitation force [1; 2]. In vibration analysis, an increase in the external force is interpreted as an increase in the internal energy, which changes the dynamic characteristics [3]. Finally, a nonlinear analysis of the structure is required. Nonlinearities are typically divided into geometric nonlinearities [4], force and displacement boundary conditions [5], and material nonlinearities [6]. The nonlinearity due to an increase in the internal energy of the general built-up structure is mostly due to a displacement boundary condition induced by the contact region inside the structure [7; 8]. The study proposes a methodology for predicting changes in the natural frequency and damping ratio when the external force increases in a built-up structure with contact surfaces. Specifically, the frequency response function (FRF) in the modal domain is derived based on the Iwan model, which is a representative constitutive model describing the nonlinearity of the dynamic characteristics due to the contact surfaces. Based on the derived FRF, the natural frequencies and damping ratios at a certain magnitude of excitation force or internal energy are predicted. The Iwan model, which was introduced by [PERSON] [9; 10], is composed of Jenkins elements of a spring-slider unit in a parallel series, where the slider represents Coulomb friction. In this model, the hysteretic behavioral effect within the structural force-displacement curve is an approximate bilinear model. [PERSON] et al. [11] extensively reviewed this model and applied it to a finite element (FE) model, which was the initial striving for Iwan model application to describe the dynamic tangential responses of lap joints. [PERSON] et al. expressed the Iwan model with four parameters in [12] and attempted to account for the dynamic behavior at the interface between the micro-slip and macro-slip regimes using the Dirac delta function. This four-parameter Iwan model is commonly used for the non-linear behavioral model of bolted joints in the micro-slip regime. In addition, a follow-up study analyzed this model in the modal domain [13], where the interface's influence on each model was considered, leading to a decoupled set of modal equations. These modal equations were a collection of parallel-series Iwan models that were used to determine the contribution of the joints to each mode. Recently, the Iwan model was used to represent built-up structures [14; 15; 16]. These studies confirmed that each mode had a different effect on the Iwan model and introduced an order reduction possibility for the full FE model of the structure. In this study, as an extension of [13], two parameters (R, \(\chi\)), out of the four introduced in [12], were estimated by modal analysis. Further, the steady state response of the Iwan model was investigated in the modal domain and the conventional modal parameter estimation method was applied to the results. The other two parameters were related to the reference position, which distinguished the micro-slip and macro-slip in the Iwan model and the backbone's curvature at the reference position. When only in the dynamic state in the micro-slip region, the structure of interest is not considered for the study as the parameters will have no effect. To estimate the change in the resonance frequency and damping ratio with respect to an increase in the external force or internal energy, the equation of motion for the built-up structure was derived by assuming that the contact surface was connected to the Iwan model.