Resonances and resonant frequencies for a class of nonlinear systems

Resonant phenomena for a class of nonlinear systems, which can be described by a SDOF model with a polynomial type nonlinear stiffness, are investigated using Nonlinear Output Frequency Response Functions (NOFRFs). The concepts of resonance and resonant frequencies are proposed for the first time for a class of nonlinear systems. The effects of damping on the resonances and resonant frequencies are also analyzed. These results produce a novel interpretation of energy transfer phenomena in this class of nonlinear systems and show how the damping effect influences the system resonant frequencies and amplitudes. The results are important for the design and fault diagnosis of mechanical systems and structures which can be described by the SDOF nonlinear model

The transmissibility of vibration isolators with a nonlinear anti-symmetric damping characteristic

In the present study, the concept of the Output Frequency Response Function (OFRF), recently proposed by the authors, is applied to theoretically investigate the transmissibility of SDOF passive vibration isolators with a nonlinear anti-symmetric damping curve. The results reveal that a nonlinear anti-symmetric damping characteristic has almost no effect on the transmissibility of SDOF vibration isolators over both low and high frequency ranges where the frequencies are much lower or higher than the isolator’s resonant frequency. On the other hand, the introduction of a nonlinear anti-symmetric damping can significantly reduce the transmissibility of the vibration isolator over the resonant frequency region. The results indicate that nonlinear vibration isolators with an anti-symmetric damping characteristic have great potential to overcome the dilemma encountered in the design of passive linear vibration isolators, that is, increasing the level of damping to reduce the transmissibility at the resonance could increase the transmissibility over the range of higher frequencies. These important theoretical conclusions are then verified by simulation studies

Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions

In this paper, the new concept of Nonlinear Output Frequency Response Functions (NOFRFs) is extended to the harmonic input case, an input-independent relationship is found between the NOFRFs and the Generalized Frequency Response Functions (GFRFs). This relationship can greatly simplify the application of the NOFRFs. Then, beginning with the demonstration that a bilinear oscillator can be approximated using a polynomial type nonlinear oscillator, the NOFRFs are used to analyze the energy transfer phenomenon of bilinear oscillators in the frequency domain. The analysis provides insight into how new frequency generation can occur using bilinear oscillators and how the sub-resonances occur for the bilinear oscillators, and reveals that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model

Preliminary design of a test rig for combining passive nonlinear isolation with active control

Resilient elements are typically used to isolate delicate equipment from a vibrating host structure. Conventionally, these isolators are designed to operate in their linear region, but more recently nonlinear isolators have been employed to increase the frequency over which vibration isolation can be achieved. Another way of improving the performance of an isolator has been to use active control in conjunction with a passive linear system. The work presented in this paper concerns the development of an experimental rig for vibration isolation and is motivated by the intention to combine the advantages of passive nonlinear isolation with active control.The structure consists of a mass suspended on four tensioned wires to form a single-degree-of-freedom system. The nonlinear stiffness of the wires is such that the system behaves like a hardening Duffing oscillator. Firstly, a static analysis is carried out, both analytically and experimentally, where the nonlinearity of the system is determined by the tension, length, cross-sectional area and Young’s modulus of the wires. For the dynamic analysis, harmonic base excitation is considered. The magnitude of the base displacement is fixed for all excitation frequencies and the level of nonlinearity is adjusted by varying the tension in the wires, a higher tension leading to a milder system nonlinearity. Finally, the motion transmissibility of the system is measured and appears to agree with the theoretical result. The rig forms a suitable platform for subsequent incorporation of an active control system for combining the benefits of passive nonlinear isolation with, for example, skyhook damping

Frequency domain energy transfer properties of bilinear oscillators under harmonic loadings

In this paper, the energy transfer phenomenon of bilinear oscillators in the frequency domain is analyzed using the new concept of Nonlinear Output Frequency Response Functions (NOFRFs). The analysis provides insight into how new frequency generation can occur using bilinear oscillators, and reveals, for the first time, that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model

Nonlinear Generalization of Den Hartog's Equal-Peak Method

This study addresses the mitigation of a nonlinear resonance of a mechanical system. In view of the narrow bandwidth of the classical linear tuned vibration absorber, a nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA), is introduced in this paper. An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is tailored according to the nonlinear restoring force of the primary system. The NLTVA parameters are then determined using a nonlinear generalization of Den Hartog's equal-peak method. The mitigation of the resonant vibrations of a Duffing oscillator is considered to illustrate the proposed developments

A new approach to nonlinear feedback control for suppressing periodic disturbances: Part 2. A Case Study

In part 1 of this paper some fundamental theoretical results for the design of a nonlinear feedback controller to suppress periodic exogenous disturbances were developed based on the frequency domain theory of nonlinear systems, and a general procedure for the controller design was proposed. In this study, Part 2 of the paper, the new approach and the theoretical results in Part 1 are demonstrated using a case study based on the design of an active vibration control system. Simulation results are given to illustrate the effectiveness of the new method and the advantage of the nonlinear feedback controller