ESTUDIO DE LA VEGETACIÓN: PLANO DE VEGETACIÓN

The model validation has been carried out using two sets of experiments. The first set of experiment will be used to identify the system and the last three sets of experiments will be used to validate the identified system. If the results of system identification for the first set agree with the results of the remaining sets of experiments then the model can be said to be validated. The system identification has been carried out in two steps. In the first step the quasi static model is used to predict the maximum force using different trial values of yield shear stress for the peak velocity of sinusoidal velocity function corresponding to the experimental results. The identified value of yield shear stress of MR fluid of the damper is the value at which the peak damper force predicted by the model, matches with the value of experimentally measured peak force. In this step the sealing force and gas spring force are also taken into account. The sealing force and the gas spring force are taken from the data sheet of RD-8040 MR damper and the off-state experimental measurements. In the second step the transient model results are superposed on the experimental results. If the simulation results are in good agreement with the experimental results the system identification can be considered to be correct.

Since the geometrical parameters of the damper were not available a priori because of being a commercial product, therefore the first step towards the validation of the model developed is to set some value of piston radius and cylinder radius such that there is a gap of 1mm. This procedure was resorted to as the supplier did not entertain the request for providing data for the flow path and fluid. The width of the piston is taken as 30 mm. These parameters are sensible because they seem to agree with the MR damper drawing of the LORD Corporation for RD8040-1 short stroke damper. Following this step the piston diameter was selected to be 30mm and the cylinder bore diameter was set at 32mm. Here it is mentioned that the MR damper may have a flow channel embedded in the piston (ref Fig 3.14) but the system identification is being carried out using the geometry of the MR damper given in Fig 3.1. In such a case also the boundary condition will not differ and this procedure can be called as equivalent system identification. This has been shown in section 3.5 of chapter 3. These values were arrived at using quasi static model to match maximum force for different values of piston and cylinder bore diameter with 1mm gap for flow of MR fluid. In this process the values of yield shear stress corresponding to the values of currents of 1A, 0.8A and 0.4 A for the above mentioned diameter of piston and cylinder bore, were found to be 16, 24 and 31 kPa respectively. The system identification was carried out using sinusoidal signal. The system identification should be done necessarily using a sinusoidal signal because a real life velocity signal is usually a continuously time varying signal and a sinusoidal signal is a close approximation to a real life signal. Using following parameters the simulation results of transient model were superposed on the experimental results for 0.34m/s amplitude and 1.786Hz sinusoidal signal. The results of the first step of system identification are given in table 4.4.

Table 4.4 Equivalent Parameters for MR damper RD8040-1

Parameter Value

Piston diameter (mm) 30

Cylinder bore diameter (mm) 32

Shear stress (kPa) at 0.4 A current 16

Gas Spring pressure (bar)

(Ref. Lord Corporation Data sheet) 2.3

The modelling results for the variation of damper force versus displacement given in Fig 4.6 show a very good agreement with the experimental results. This leads to the confirmation that the yield shear stress value of 16 kPa for 0.4 A current is correctly identified. The simulation results for the variation of damper force with velocity have been given in Fig 4.7. The modelling results and the experimental compare very well with each other. The mean error in the comparison of modelling results with the experimental results has been found to be 5.6 %. The peak damper force predicted by the model is in excellent agreement with the experimental results. Using the results of quasi static model and transient model the sealing friction force has been found to be 50N. The force due to gas spring has been found to be 30N. If the data for the gas spring pressure given in the table 4.4 is used the gas spring force is given as 28N. The difference in gas pressure obtained from the experimental measurements and manufacturer data differ by 7%. Since the major component of error is attributable to manufacturing tolerance in gas spring pressure as compared to the experimental error therefore the value of gas spring force is considered acceptable.

Fig 4.6 Variation of force versus displacement for 0.4A current (yield shear stress of 16kPa)and 1.786Hz sinusoidal velocity input for experimental measurement and model simulation.

Fig 4.7 Variation of force versus velocity for 0.4A current (yield shear stress of 16kPa)and 1.786Hz sinusoidal velocity input for experimental measurement and model simulation.

The next experiment was performed at the 0.8A excitation current for the MR damper keeping all the other parameters given in the table 4.4 as same. The value of yield shear stress corresponding to 0.8A has been found to be 24 kPa. The damper rig was driven with 2.5Hz sinusoidal velocity signal. The results for the variation of damper force with dispalcement and damper force with velocity are given in Fig 4.8 and 4.9 respectively. In this case also the model results are in very good agreement with the experimental results. Therefore the shear stress value 24 kPa for the MR fluid of the damper has been correctly identified.

Fig 4.8 Variation of force versus displacement for 0.8A current (yield shear stress of 24 kPa)and 2.5 Hz sinusoidal velocity input for experimental measurement and model simulation.

Fig 4.9 Variation of force versus velocity for 0.8A current (yield shear stress of 24 kPa)and 2.5 Hz sinusoidal velocity input for experimental measurement and model simulation.

The third experiment for 1A excitation current for MR damper, was carried out using a sinusoidal velocity signal of 0.523Hz frequency. The yield shear stress corresponding to 1A exciation current was found to be 31 kPa. The results for the variation of damper force with velocity and damper force with displacement are given in Fig 4.10 and 4.11. The experimental results and the results of transient modela are also in close agreement with each other. Since the modelling results and the experimental results are in close agreement with each other for all the three experiments therefore the model parameters can be said to be correctly identified. The system identfication also establishes the relationship between the excitation current and the

yield shear stress. The variation yield shear stress with applied current for the MR damper is given in Fig 4.12.

Fig 4.10 Variation of force versus displacement for 1A current (yield shear stress of 31 kPa)and 0.523 Hz sinusoidal velocity input for experimental measurement and model

simulation.

Fig 4.11 Variation of force versus velocity for 1A current (yield shear stress of 31 kPa) and 0.523 Hz sinusoidal velocity input for experimental measurement and model simulation.

The equation for the yield shear stress as a function of excitation current is given as follows: c c i i i 104375 72833 63541 3 ₀2 0     (4.2)

The second set of experiments were also performed using sinusoidal velocity signal. The first experiment of the second set was performed using 1.25Hz sinusoidal velocity signal.The results for the variation of damper force with dispalcement have been shown in Fig 4.13.

Fig 4.13 Variation of force versus displacement for 0.4A current (yield shear stress of 16 kPa)and 1.25Hz sinusoidal velocity signal.

Fig 4.14 Variation of force versus velocity for 0.4A current (yield shear stress of 16 kPa) and 1.25Hz sinusoidal velocity signal.

The variation of damper force with velocity has been shown in Fig 4.14. These results also confirm that the transient model is able to predict the damper force within 5-7% mean error. The next experiment was performed using the vellocity signal of same frequency as the previous one. The damper current was changed to 0.8 A. If the Fig 4.14 and 4.16 are compared with each other it will be noticed that an increase in yield shear stress due to the increase in current leads to an increase in the area of the hysteresis loop. The shape of the hysteresis loop also changes. This is because an increase in yield shear stress results in an increase in plug thickness. Due to the increase in plug thickness the effect of body force is more significant. Similarly the effect of transient boundary condition also becomes significant and swelling of the hystresis loop is also because of transient effects listed in the previous chapter. Curve for the variation of force with displacement is, by and large, symmetric when the damper current was kept at 0.4A. In case of the second experiment with 0.8A current, the force versus

Fig 4.15 Variation of force versus displacement for 0.8A current(yield shear stress of 24 kPa) and 1.25Hz sinusoidal velocity signal.

Fig 4.16 Variation of force versus displacement for 0.8A current(yield shear stress of 24 kPa) and 1.25Hz sinusoidal velocity signal.

There were two more experiments conducted with sinusoidal velocity signal of 2.5Hz and 0.56 Hz velocity signals. These experiments were performed for 0.8A and 1A excitation current respectively. The outcome of the comparison of modelling results with the experimental results remains the same as has been discussed previously. At this stage it will be interesting to reexamine all the force velocity curves and discuss the interpretation of the results. If the region of the force velocity curves corresponding to zero or negligible force are examined it will be noticed that for same values of damper velocities the damper force is zero during the up and down strokes of the MR damper. These regions correspond to the operational conditions when the pressure gradient across the flow path is negligibly small. In such a case the flow in the damper channel is purely piston driven flow or a Couette flow. When the piston velocity increases from zero value to the value corresponding to zero damper force, the damper force and damper velocity have opposite sign. In this velocity range the damper either tends to approach the end of the stroke or it just starts the reverse stroke. The pressure gardient in this region is governed by the body forces and the effect of fluid inertia. It can be seen that as the frequency of the velocity signal increases the velocity range for boundary driven flow increases. The combination of low frequency and high yield shear stress results in a significant reduction the range of velocities for which the damper force and damper velocity have opposite sign. If there are significant amount of air bubbles in the damper fluid, the fluid can become more compliant and the above mentioned velocity range can further increase. The results of Sims et al (2000), Batterbee et al (2007) and Wang and Gordaninejad (2007) point to some similar possibility. In Wang and Gordaninejad (2007) it has been mentioned that the presence of air bubbles can reduce the bulk modulus of an MR fluid by two orders of magnitude. This observation is important from the design and manufacturing point of view, because the presence of air bubbles in an MR damper can significantly affect the damper force. Therefore, a

fluid. Therefore the force velocity curves can give a good indication of the presence of gas bubbles in an MR damper fluid.

Fig 4.17 Variation of force versus displacement for 1A current(yield shear stress of 31 kPa) and 0.56 Hz sinusoidal velocity signal.

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Fig 4.18 Variation of force versus velocity for 1A current(yield shear stress of 31 kPa) and 0.56 Hz sinusoidal velocity signal.