8.1 Conclusions
The goal of this thesis has been to model the quasi-static and dynamic behavior of a wire rope spring in the tension-compression mode. In order to achieve this goal a number of preliminary steps had to be taken.
The wire rope spring exhibits asymmetric hysteretic behavior with softening behavior in compression and hardening stiffness in tension. For small amplitudes the wire rope spring possesses softening behavior which via quasi-linear behavior gradually changes into hardening behavior for increasing amplitudes. Furthermore, hardening overlap in tension has been found. Some typical nonlinear characteristics have been found for the dynamic behavior: an amplitude dependent resonance frequency and various superharmonic resonances. However, wire rope springs are suitable for damping vibrations with frequencies higher than 15 Hz if a wire rope spring is loaded with a dummy mass of 50 kg. Due to the resonance frequency the amplitude of vibrations below 15 Hz is enlarged. Furthermore, shock experiments have confirmed that the wire rope springs are very good shock absorbers.
A modified version of Bouc-Wen model has been used to describe the behavior of the wire rope spring because the Bouc-Wen model itself cannot represent soft-hardening and hardening overlap. Although the Bouc-Wen model is a commonly used model to describe hysteretic behavior since it can represent softening, hardening and quasi-linear behavior, it is not an easily applicable model. First, it is really difficult to understand the physical meaning of some of the parameters. Secondiy, a redundant parameter phenomenon is present since different values for
P
and y can lead to almost identical hysteretic curves. Two possible solutions can avoid this redundancy: (1) adding a constraint for ,O and y restricts the possible combinations drastically or (2) adopting an altered version of the Bouc-Wen model in which the parameter y is omitted. However, these scenarios for ,L3 and y do not yield the same dynamic response. An explanation for this has not been found yet. The dynamic response of the Bouc-Wen model exhibits no subharmonic responses except for large values of n.The identification of all parameters has been a difficult task, because the weight on the various parameters varies a lot during the process. Eventually, a three stage identification strategy has led to satisfying results. The model response and the experimental results are in good agreement, but for small amplitude levels still a difference occurs.
The shock simulations and the experimental shock responses agree qualitatively well.
However, the amplitude of the vibration differs. The flexibility and the damping of the frame,
70 8.2 Recommendations for further research
which has been needed to put the total system on the shock table, may be responsible for this, because the excitation, which is also used in the simulations, has been measured directly on the table. The theoretical and experimental frequencies of the resulting free vibration are in good agreement.
Frequency sweeping has provided a first impression about the vibration behavior of the modified Bouc-Wen model with the identified set of parameters. Qualitatively the model response ar,d the experimental results show good agreement. For 'larger amplitudes both respefises reserLb!e each other x,*,*~e!! in 2 qi~~~t,it&,tix~ xxmv " -J ) althmigh
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it epm-s th& model response is damped a Sit weaker since the number of superharmonic resonances is larger. For small amplitudes both responses quantitatively differ since the model response is damped too heavily. The shooting method in combination with path following has been used to calculate periodic solutions of the modified Bow-Wen model. The local stability has been determined using Floquet theory. By decreasing the tolerance of the integration scheme all solutions are stable.For small amplitude levels, both for quasi-static as well as for dynamic behavior, the wire rope spring exhibits a behavior which differs significantly from the behavior for larger am- plitude levels. Hence, it seems not possible to describe quantitatively the behavior for all amplitude levels with one set of parameters. The quasi-static experiments and the identifica- tion method are routines which have to be implemented in the design process to calculate the response of the wire rope spring system in a certain application or environment. If a priori it is known that, the wire rope spring is loaded with_ large amplitude or small amplitude levels only, the described method will likely lead to good results and will be useful in the design process.
8.2 Recommendations for further research
For further research it is recommended to analyse the small amplitude level behavior of the wire rope spring in more detail since it is yet not totally understood what happens. It cannot be expected that the modified Bouc-Wen model will be able to describe the behavior for all amplitude levels with one set of parameters. Possible solutions are to identify a second set of parameters for small amplitudes only or to try to establish a amplitude dependent relationship for one or more parameters. The latter will be a very difficult task and it is not known whether this yields better results. A third solution might be to develop a model on physical grounds, although it can be questioned whether this is feasibie.
The Bouc-Wen model needs to be investigated more in detail as well, because the inter- action between ,Ll and y still gives rise to various questions. The main issue might be why two scenarios for ,Ll and y lead to almost identical hysteretic curves but do not yield identical dynamic responses. Additionally the Ozdemir model has to be addressed as well, because omitting the parameter y means that the redundant parameter phenomenon will vanish too.
During this master thesis project only the behavior in the tension-compression mode has been considered. Since wire rope springs are used in the shear and roll mode as well, it is very logical to analyze the behavior in these modes. A more symmetric hysteretic behavior can be expected.
Eventually, it will be the challenge to analyze the behavior of the wire rope spring when the motion is not restricted to one mode only. This will certainly be more comparable to the real world because it can never be guaranteed that the excitation is in one direction only. A
consequence seems that interaction between the different directions will play a role, leading to a serious increase of the calculation time, which really can be a disadvantage. However, if in all directions the model yields good results the calculation tool can be implemented into the design process and it will be possible to calculate beforehand the behavior of the spring, that will be used, to find out whether it is suitable for the task.
72 8.2 Recommendations for further research