Numerical and experimental studies of a nonlinear vibration system

Abstract

The objective of this research is to show that nonlinearity can be used to improve vibration absorption and suppression of unwanted vibrations in a main system due to external excitation. This was shown by investigating two systems a SDOF (with hardening nonlinearity) and a 2DOF (with softening nonlinearity). The aim of carrying out these investigations was to introduce a passive nonlinear system that can update itself and self-regulate to suppress undesired oscillations. To fulfil the desired gaol, various types of springs were considered and investigated. A commercially available spring called Mag-spring has been chosen and a function for its nonlinearity has been investigated. Mag-Spring is a newly invented spring which is designed to exhibit constant force at its operating range. However, this spring has a special non-linear behaviour before reaching to the constant force domain which is the main focus of the investigation presented in this thesis. The nonlinear behaviour of Mag-spring encouraged the idea that vibration design is possible by the advantages that can be gained from magnetic technology. The added benefit through this new Mag-spring, is that it solved some of the concerns assotiated with old available ones. The most concern norrowing the usage of magnetic springs, is the air gap between the two magnets, which make the attraction or repulsive force unstable through the spring’s working range. Linmot Company, introduced a solution to this concern by introducing a teflone that works as a bearing between the two magnets, which fixed the distance between them while they are sliding againest each other. In the first scenario of this study, a hardening nonlinear spring was added in parallel to a system with a single degree of freedom. The system will remain single degree of freedom as the spring was added in parallel without additional mass. The hardening spring shows low stiffness at low amplitude and high stiffness at higher amplitude. In this study, it was shown that nonlinearity affects the dynamic performance of a system and makes the natural frequency amplitude dependant. As the amplitude of vibration increases, consequently, stiffness increases and the natural frequency shifts away from the excitation frequency. For this investigation, a vibrating system with one degree of freedom has been built based on a mathematical model simulated and tested in Matlab software. Mag-spring was used to introduce the nonlinear stiffness to the system. Unbalance mass mounted to a disc fixed to a rotational machine has been used to create a forced vibration system with variable frequency. The response of the system with and without nonlinearity effect was monitored with an accelerometer. Simulation and experimental results showed that nonlinearity could shift the resonance frequency of the SDOF system by 10% (hardening of the system), without affecting the stiffness of the system at normal working condition. In the second scenario, a softening nonlinear spring was added as a vibration absorber to a system with a single degree of freedom, to make the system with two degree of freedom. The softening spring shows high stiffness at low amplitude and low stiffness at high amplitude. The rationale behind this is to introduce a spring which is hard at high frequency and soft at low frequency, which as a result will make the ratio √(k_a/m_a ) of the absorber follows the excitation frequency (ω) allowing the system to update itself and self-regulate providing vibration cancellation at more than one frequency value and widen the vibration cancellation range (ω_n2-ω_n1). It was shown that the Mag-spring could show a softening behaviour in a limited domain if its operating position is shifted. A program has been written to simulate the behaviour of all nonlinear system with two degree of freedom (nonlinear absorber). At this program, the maximum amplitude of each time domain was used to produce the frequency domain of the amplitude of the system. The amplitude of the vibration for a linear and a nonlinear absorber was compared. The results showed that the nonlinear absorber suppresses and reduces the vibration amplitude of the main system better than the linear absorbers with up to 60% reduction in magnification ratio and from 5% to 10% in widening the cancellation range (ω_n2-ω_n1). In the last scenario of this study, 4 different ideal softening stiffness curves were introduced based on theoretical methods. Their vibration response was calculated and compared to the nonlinear absorber (Mag-spring) and a linear absorber. This study shows that when nonlinearity is designed properly, it could provide a distinguished vibration cancellation response resulting more than 60% vibration cancellation improvement. This study demonstrated the possibility of developing a passive self-regulating tuned mass system involving the usage of nonlinearity. Nonlinearity will enhance the vibration cancellation by allowing the system to update itself and as a result will make the vibration absorption to be effective within a frequency range rather than single frequency unlike the classical tuned mass system. This study, to the best knowledge of the author, can be classified as an uncommon study in vibration systems investigations.