Bilinear amplitude approximation for piecewise-linear oscillators

Currently, the interest in monitoring the vibration of dynamical systems has been increasing. For example, large and complex air and space structures include vibration monitoring systems to forecast sudden failures. Such vibration monitoring is used to diagnose structural health by analyzing vibration characteristics (such as frequency and amplitude). Among the vibration characteristics used, the vibration amplitude is especially important because it can directly affect the life of the system. Finite element (FE) models are often used to analyze vibration characteristics. For low dimensional systems, full FE models can be used. For high dimensional systems, the computational cost of analyzing full FE models can often be prohibitive. To circumvent this difficulty, many methods for creating reduced-order models (ROMs) have been developed for various systems [1–14], with the majority being focused on linear systems. ROMs for linear systems can be efficiently constructed by using approaches based on linear transformations [15, 16] such as component mode synthesis (CMS) [17]. However, constructing ROMs for systems with piecewise-linear nonlinearity (caused for example by intermittent contact) require careful treatment. Accurate ROMs can be constructed for such systems using linear transformations [18–20], or using nonlinear normal modes [21–23]. Recently, Saito et al. [24] developed a reduced-order modeling method based on bilinear modes (BLMs) for dynamical systems with piecewise-linear nonlinearity. They observed that the space spanned by the most dominant proper orthogonal modes (POMs) of a system is also spanned by a set of linear normal modes for the system with special boundary conditions at the surface where the intermittent contact takes place. The special modes were referred to as BLMs. Hence, the most dominant POMs are well approximated by linear combinations of BLMs. Thus, ROMs based on BLMs are accurate and have a low dimension. Nonetheless, predicting the vibration amplitude requires the calculation of the nonlinear forced response of the ROMs. Namely, the nonlinear forced responses have to be obtained by direct numerical calculation (e.g., by using a variable step Runge-Kutta method), which incurs a large computational cost despite the fact that the ROMs are low dimensional. For example, mistuned bladed disks with cracks have piecewise-linear nonlinearity due to the intermittent contact at the crack surfaces. Therefore, to obtain the amplitude of vibration at the resonant frequencies, nonlinear forced responses need to be calculated using more efficient numerical methods (e.g., hybrid frequency/time domain methods) [25–28]. In this paper, a novel technique to approximate the vibration amplitude at the resonant frequencies of dynamical systems with piecewise-linear nonlinearity is proposed. Here, it is assumed that the forcing applied to the system is harmonic and the response of the system is periodic. Thus, quasi-periodic or chaotic dynamics are not considered. The proposed technique is referred to as bilinear amplitude approximation (BAA). BAA constructs approximations for the periodic steady-state response of the system at resonant frequencies. For example, consider that a structure has a crack which opens and closes during each vibration cycle. BAA uses linear modes (similar to BLMs) from two different systems: one with an open crack and the other where there is sliding at the crack surfaces. By doing so, BAA does not require the numerical integration of nonlinear ROMs to calculate the vibration amplitude at resonant frequencies. Consequently, large savings in computational costs are obtained.