Vibration Isolation of the LEGO® Plate
Like we see in the post
http://www.openadaptronik.de/2017/01/09/modal-analysis-of-the-lego-plate/
the LEGO® plate can be considered within a certain frequency as it has a linear behaviour. The test must be done again to get the modes of the system because the boundary conditions are changed. In fact, in the previous post, the plate was suspended but now the structure is joined together with the actuators.
A service structure is used to be sure that the incoming disturbance from the ground can be neglected thus the structure can be analysed in a proper way. The structure will be measured in the corner points illustrated in the figure.
With the same procedure used before the eigenfrequencies and adimensional damping rations are calculated. A modal reconstruction of the system is done by means of a least square estimation and the reconstructed response are shown in the figures below. Where: Co-Located means hammer in point 1 and response in point 1 (equivalently 2 in 2 and so on), Short-Side means hammer in 1 and response in 2 (equivalently hammer in 2 response in 1 or hammer in 3 and response in 4 and so on), Long-Side means hammer in 1 and response in 3 and so on, Cross-Response means hammer in 1 and response in 4 and so on.
It is thus possible to have the data form the plate needed in terms of frequency damping and mode shapes:
The implementation of the control is done following a modal approach synthetized below:
Where:
Good simulation results are achieved with the control just defined and they can be compared with the ideal case in which every mode of the system is controlled autonomously.
In the real system implementation, a careful choice of the position of the accelerometers must be done. In fact, it is possible to see from the open loop transfer function of the system from the input voltage to the sensor output that, around 150 Hz, an internal resonance, not seen before, appear lowering the phase and making the control no more co-located. This can cause instability in the control loop. The problem can be solved positioning the sensor below the plate. The effect of this choice can be seen from the red bounded region in the figure below.
Good experimental results can be achieved and they are reasonably equal to the predicted ones. Only the response of point 3 is shown as example.
The control logic must be as simple as possible and for this reason the control matrix used until now, centralized control, is simplified to a decentralized control following the rule:
In this way, it is possible to maintain the same effect of the control on the first mode. This solution lead to good attenuation also for what concerns the second and the third mode. Experimental results of the decentralized control are shown as before in point 3 as example.
The system is very well damped in both the control logic tried in the frequency region of interest and a decentralized control con produce the same effect of the centralized version.
Until now the system is controlled via a sampling frequency of 10 kHz. This is definitively too high for an implementation with low-cost hardware, thus experiments with 1 kHz and 0.5 kHz are carried out.
As it possible to see a reduction in the attenuation is evident. In terms of gains that is possible to give to the system the reduction needed are:
Where:
Decentralized control can act in a better way lowering the sampling frequency. A good way to overcome the sampling problem is to implement the decentralized control in an analogue way.
More detailed explanation of the control implementation and experimental set up can be found on slide share here:
https://www.slideshare.net/OpenAdaptronik/vibration-isolation-progect-legor
It is proved, in any case, that vibration isolation can be carried out with the usage of low cost actuators and a simple control logic.
Good bye,
Rocco.
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