Calculation of Nonlinear Systems Under Narrow Band Excitation Using Equivalent Linearization and Path Continuation

Turbomachines experience a wide range of different types of excitation during operation. On the structural mechanics side, periodic or even harmonic excitations are usually assumed. For this type of excitation there are a variety of methods, both for linear and nonlinear systems. Stochastic excitation, whether in the form of Gaussian white noise or narrow band excitation, is rarely considered. As in the deterministic case, the calculations of the vibrational behavior due to stochastic excitations are even more complicated by nonlinearities, which can either be unintentionally present in the system or can be used intentionally for vibration mitigation. Regardless the origin of the nonlinearity, there are some methods in the literature, which are suitable for the calculation of the vibration response of nonlinear systems under random excitation. In this paper, the method of equivalent linearization is used to determine a linear equivalent system, whose response can be calculated instead of the one of the nonlinear system. The method is applied to different multi-degree of freedom nonlinear systems that experience narrow band random excitation, including an academic turbine blade model. In order to identify multiple and possibly ambiguous solutions, an efficient procedure is shown to integrate the mentioned method into a path continuation scheme. With this approach, it is possible to track jump phenomena or the influence of parameter variations even in case of narrow band excitation. The results of the performed calculations are the stochastic moments, i.e. mean value and variance.

Optimal Design of Tuned Mass Damper Inerter for Base-Isolated Buildings

Tuned mass dampers (TMD) are installed in base-isolated building to suppress the excessive isolator displacement and acceleration responses of primary structure. By incorporating an inerter element into the original configuration, the seismic performance of TMD is significantly enhanced. In this work, optimal solutions of tuned mass damper inerter (TMDI) for improving the seismic resilience of base-isolated building are proposed. The analytical formulations of optimal design of TMDI are respectively developed to minimize the H2 norm of the displacement of primary structure relative to the base floor and the isolator displacement. The performance of presented optimal methods are validated by using stationary responses under the stochastic excitations. Additionally, the seismic performance of TMDI with parameters obtained from the proposed method are compared with the established methods.

On representing noise by deterministic excitations for interpreting the stochastic resonance phenomenon

Adding noise to a system can ‘improve’ its dynamic behaviour, for example, it can increase its response or signal-to-noise ratio. The corresponding phenomenon, called stochastic resonance, has found numerous applications in physics, neuroscience, biology, medicine and mechanics. Replacing stochastic excitations with high-frequency ones was shown to be a viable approach to analysing several linear and nonlinear dynamic systems. For these systems, the influence of the stochastic and high-frequency excitations appears to be qualitatively similar. The present paper concerns the discussion of the applicability of this ‘deterministic’ approach to stochastic systems. First, the conventional nonlinear bi-stable system is briefly revisited. Then dynamical systems with multiplicative noise are considered and the validity of replacing stochastic excitations with deterministic ones for such systems is discussed. Finally, we study oscillatory systems with nonlinear damping and analyse the effects of stochastic and deterministic excitations on such systems. This article is part of the theme issue ‘Vibrational and stochastic resonance in driven nonlinear systems (part 1)’.