In this paper the backbone curves of a two-degree-of-freedom nonlinear oscillator are used to interpret its behaviour when subjected to external forcing. The backbone curves describe the loci of ...dynamic responses of a system when unforced and undamped, and are represented in the frequency–amplitude projection. In this study we provide an analytical method for relating the backbone curves, found using the second-order normal form technique, to the forced responses. This is achieved using an energy-based analysis to predict the resonant crossing points between the forced responses and the backbone curves. This approach is applied to an example system subjected to two different forcing cases: one in which the forcing is applied directly to an underlying linear mode and the other subjected to forcing in both linear modes. Additionally, a method for assessing the accuracy of the prediction of the resonant crossing points is then introduced, and these predictions are then compared to responses found using numerical continuation.
The High Static Low Dynamic Stiffness (HSLDS) concept is a design strategy for a nonlinear anti-vibration mount that seeks to increase isolation by lowering the natural frequency of the mount whilst ...maintaining the same static load bearing capacity. It has previously been proposed that an HSLDS mount could be implemented by connecting linear springs in parallel with the transverse flexure of a composite bistable plate — a plate that has two stable shapes between which it may snap. Using a bistable plate in this way will lead to lightweight and efficient designs of HSLDS mounts. This paper experimentally demonstrates the feasibility of this idea. Firstly, the quasi-static force–displacement curve of a mounted bistable plate is determined experimentally. Then the dynamic response of a nonlinear mass–spring system incorporating this plate is measured. Excellent agreement is obtained when compared to theoretical predictions based on the measured force–displacement curve, and the system shows a greater isolation region and a lower peak response to base excitation than the equivalent linear system.
•We propose the use of a tuned-inerter-damper (TID) for vibration suppression in cables.•The TID performance is analysed in parallel to the performance of viscous dampers.•A new procedure for TID ...systems optimisation is presented.•We develop a practical design method for TIDs and dampers, based on contour plots.•The improved performance of TIDs is confirmed through numerical simulations.
This paper considers the use of a tuned inerter damper (TID) system for suppressing unwanted cable vibrations. The TID consists of an inerter, a device that exerts a force proportional to relative acceleration, coupled with a parallel spring and damper. It may be thought of as similar to a tuned-mass-damper, but requiring two terminals. As a two terminal device, the performance of the TID is compared to the classical use of viscous dampers (VD) located close to one of the cable supports. We show that the limitation that exists with VDs, where the modal damping of the cable cannot exceed a maximum level for a given damper location, can be overcome through the use of the TID at the same location. A practical design methodology, based on the minimisation of the displacement amplitude at the mid-span of the cable subjected to support excitation, is proposed. An example where a cable is subjected to the El-Centro earthquake demonstrates that the system’s response is improved when a TID is used instead of a VD.
•First demonstration of control-based continuation on a MDOF nonlinear system.•Systematic experimental characterisation of nonlinear dynamic features, including isola.•Method complexity scales with ...instability dimension, rather than system’s dimension.•Control-based continuation does not necessarily require complex control strategies.•Higher-harmonics in the applied force affect the system’s stability properties.
This paper presents a systematic method for exploring the nonlinear dynamics of multi-degree-of-freedom (MDOF) physical experiments. To illustrate the power of this method, known as control-based continuation (CBC), it is applied to a nonlinear beam structure that exhibits a strong 3:1 modal coupling between its first two bending modes. CBC is able to extract a range of dynamical features, including an isola, directly from the experiment without recourse to model fitting or other indirect data-processing methods.
Previously, CBC has only been applied to (essentially) single-degree-of-freedom (SDOF) experiments. This is the first experimental demonstration of CBC in the presence of complex MDOF response features such as internal resonance, isola, and Neimark-Sacker bifurcations. In this paper we show that the feedback-control methods and path-following techniques used in a SDOF context can equally be applied to MDOF systems. A low-level broadband excitation is initially applied to the experiment to obtain the requisite information for controller design and, subsequently, the physical experiment is treated as a “black box” that is probed using CBC. The invasiveness of the controller used is analysed and experimental results are validated with open-loop measurements. Good agreement between open- and closed-loop results is achieved, though it is found that care needs to be taken in dealing with the presence of higher-harmonics in the force applied to the structure.
Control-based continuation is a recently developed approach for testing nonlinear dynamic systems in a controlled manner and exploring their dynamic features as system parameters are varied. In this ...paper, control-based continuation is adapted to follow the locus where system response and excitation are in quadrature, extracting the backbone curve of the underlying conservative system. The method is applied to a single-degree-of-freedom oscillator under base excitation, and the results are compared with the standard resonant-decay method.
The high static low dynamic stiffness (HSLDS) concept is a design strategy for an anti-vibration mount that seeks to increase isolation by lowering the natural frequency of the mount, whilst ...maintaining the same static load bearing capacity. Previous studies have successfully analysed many features of the response by modelling the concept as a Duffing oscillator. This study extends the previous findings by characterising the HSLDS model in terms of two simple parameters. A fifth-order polynomial model allows us to explore the effects of these parameters. We analyse the steady-state response, showing that simple changes to the shape of the force displacement curve can have large effects on the amplitude and frequency of peak response, and can even lead to unbounded response at certain levels of excitation. Harmonics of the fundamental response are also analysed, and it is shown that they are unlikely to pose significant design limitations. Predictions compare well to simulation results.
In an attempt to improve the performance of vibration-based energy harvesters, many authors suggest that nonlinearities can be exploited to increase the bandwidths of linear devices. Nevertheless, ...the complex dependence of the response upon the input excitation has made a realistic comparison of linear harvesters with nonlinear energy harvesters challenging. In a previous work it has been demonstrated that for a given frequency of excitation, it is possible to achieve the same maximum power for a nonlinear harvester as that for a linear harvester, provided that the resistance and the linear stiffness of both are optimized. This work focuses on the bandwidths of linear and nonlinear harvesters and shows which device is more suitable for harvesting energy from vibrations. The work considers different levels of excitation as well as different frequencies of excitation. In addition, the effect of the mechanical damping of the oscillator on the power bandwidth is shown for both the linear and nonlinear cases.
There is an increasing desire in the railway industry to improve the longevity of wheels and rails without reducing performance in other ways (e.g. worsening passenger comfort). One way of reducing ...track and wheel wear is to reduce the primary yaw stiffness, significantly diminishing the costs associated with maintenance and emergency repairs, resulting however in reduced passenger comfort and high-speed stability. This paper, using a two-axle railway vehicle case study, demonstrates the potential of using passive, inerter-based suspensions to concurrently improve ride comfort and reduce track wear. The industrial parameter
is used to quantify the frictional energy lost at the contact patch under curving conditions, and the lateral RMS carbody acceleration is used to quantify passenger comfort under straight running conditions, with lateral track disturbances taken from real track data. Optimisation results conclude that, with the default yaw stiffness value, compared with the default spring-damper configuration in the primary lateral suspension, employing beneficial inerter-based configurations can improve passenger comfort by up to 43%. If the yaw stiffness is reduced such that the track wear is improved, similar improvements in passenger comfort can still be achieved with lateral inerter-based suspensions; for example, an improvement of 33% can still be achieved with a 50% reduction in yaw stiffness. Furthermore, when an inerter-based lateral suspension is used together with a Hall-Bush longitudinal suspension, the passenger comfort rises to 40%, which relates to a 25% improvement when compared with a non-inerter lateral plus Hall-Bush longitudinal setup.
This paper considers the dynamic response of coupled, forced and lightly damped nonlinear oscillators with two degree-of-freedom. For these systems, backbone curves define the resonant peaks in the ...frequency–displacement plane and give valuable information on the prediction of the frequency response of the system. Previously, it has been shown that bifurcations can occur in the backbone curves. In this paper, we present an analytical method enabling the identification of the conditions under which such bifurcations occur. The method, based on second-order nonlinear normal forms, is also able to provide information on the nature of the bifurcations and how they affect the characteristics of the response. This approach is applied to a two-degree-of-freedom mass, spring, damper system with cubic hardening springs. We use the second-order normal form method to transform the system coordinates and identify which parameter values will lead to resonant interactions and bifurcations of the backbone curves. Furthermore, the relationship between the backbone curves and the complex dynamics of the forced system is shown.
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