3. Nonlinear beam dynamics
3.5 Conclusions
In this chapter, finally a real nonlinear system is taken into account. The majority of the nonlinear properties highlighted are visible thanks to the frequency content analyses. In particular the wavelet method is much more precise and able to catch instantaneous phenomena at relatively high frequency than the spectrograms owing to the decreasing time windowing needed to have a proper resolution as the frequency grows.
Also there is an exceptional correspondence of the experimental and numerical values of the frequencies, which lends the work a certain prestige.
Conclusions
Chapter § 1 of the work is the perfect representation of how difficult can be to obtain a satisfying result when one has to deal with discretising complex tridimensional automotive structures. In this specific case, many routines affected the final results:
• PrismaticDiscretisation.m, where the nodes of an equispaced grid with different distances in the three dimension was created. In particular, the new routine is faster than the previous one;
• Fea_simpler_Prismatic.m, where many of the generated nodes were deleted and only the ones near to the initial SW mesh survive. Many attempts to choose properly the “correction factor” were carried out and the best results came with the value of 1;
• Hex_Mat_assign.m, where the right material properties were assigned to the surviving Hexs on the base of the material of the nearer Tets. The best way to treat the uncoupled Hexs is to attribute them the most diffuse features of the surrounding cuboid that have the material correctly assigned.
Afterwards, in § 2, an overview of the foremost aspects of nonlinear systems was presented. The presence of the superharmonics of the fundamental frequency in their frequency content is surely one of them. Another important nonlinear characteristic is the hardening/softening phenomenon, i.e.
the increase/decrease of the values of the natural frequencies happens when the amplitude of the oscillations grows.
In order to better understand these properties various SDOF damped and undamped system were simulated in Simulink. The most complete equation of motion for this system is:
2 3
1 2 3 0
mx cx k x k x+ + + +k x = (C)
The effects of variating one parameter at time were analysed. Due to the fact that k2 is always negative, instability problems could occur if the system was excited with a too intense IC.
Moreover, it resulted that high damping made more complicated to spot nonlinear symptoms for the used system.
The most appropriate tools used to deepen nonlinear system are:
• Spectrograms: that are based on the FFT, representing structures with summation of sines and cosines. Certainly, a crucial parameter to be set concerning this method is the frequency resolution: the higher it is, the more efficient one is concerning the individuation of the precise values of the frequency contents. Vice versa, the advantage of having a sufficient low resolution is the increasing probability to identify more instantaneous phenomena;
• Wavelet, which employs waves of various shapes and now are widespread in the word of compressing sounds. Here the consequences of varying an input parameter in MATLAB is way minor with respect to the former.
The difference between the two tools is that the spectrograms have a constant delta-time sample Δt and only captures a mean value of the instantaneous frequencies and therefore are less able to catch instantaneous phenomena than the wavelet method. Indeed, the fact that its resolution increases for higher frequencies, does make it more suitable for nonlinear systems.
The theoretical concepts of § 2 were eventually observed in two real experiment campaigns in chapter § 3, where a vibrating beam with various sources of nonlinearity. Unfortunately, the acquirements of the 2nd campaign had been marked with an unwanted low-pass filter that cancelled
the frequency contents of the test over 40÷45 Hz. Nevertheless, many nonlinear symptoms could be highlighted.
The first analysed one was the softening effect owed to the increase of the oscillation period when the hypothesis of little oscillations could not be done, that had been spotted in Figure 3.3.1.7.
Afterwards, the softening and hardening effects noticeable respectively with magnets in attraction and repulsion. Using a test of the first campaign also a validation of what (2.2.2.6) states, i.e. the fact that linearising the expression of a magnet force up to the 3rd order is completely enough to fully reproduce its behaviour. In order to make this, Figures 3.3.3.10-11 were produced to demonstrate that the maximum number of visible superharmonics of the fundamental is 3.
As regards the test involving the non-holonomic constraint, the wavelet method was able to precisely correlate the resonance frequencies calculated using the Lupos simulation tool to the ones visible in the frequency contents. In particular, the first 3 frequencies related to the xz bending modes of the free cantilever beam are clearly visible in Figure 3.3.3.5. The fundamental one was not visible with the value of 2.97 Hz (Table 3.4.3), but with a weighted mean with the 1st natural frequency of the configurations with a clamped-pinned beam (setup 3) that considered the time duration of a complete oscillation and the instants of contact during the oscillation itself. This had been documented with the help of Figure 3.3.3.6.
At the very end of the thesis, the details of the simulation of the systems were reported. It can be said that the frequency contents of the tests were analysed with the use of the final Table of the work, i.e. Table 3.4.3, whose topic is the list of the first 4 natural frequencies of system in configurations 3 and 4 concerning the xz bending modes.
Reference
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