Luận án Study on the quasi-zero stiffness vibration isolation system

Nội dung text: Luận án Study on the quasi-zero stiffness vibration isolation system

1. 102 Stable branch vˆ ac1 Unstable branch 101 | F |T 100 10-1 10-1 100 101 102 (rad/s) ˆ Fig. 5.37. Force transmissibility of the QSAVIM using PC for various values of Vac1 including 7.89; 14.46, 18.41 and 24.99; 39.45; the same other parameters as in Fig.5.36. The effect of nonlinearity in dynamic characteristic, the amplitude-frequency curve can be bended as analyzed above, which indicates the appearance of the down (JD) and ˆ up (JU) jump frequency as expressed in Fig.5.38. Herein, the values of µ and Vac1are given in left-top corner panel, the other parameters are the same as in Fig.5.35. The points on the response curve lying between JD and JU are unstable solutions because these points are in the unstable region calculated by Eq. (5.76). When the down jump frequency which is also the peak frequency p moves closely to zero, the phenomenon of the frequency jump may be neglected, meaning that all points on the amplitude- frequency curve are stable as numerically simulated by the dashed line in Fig.5.39. ˆ Likewise, the stable branches of the proposed model with various values of µ and Vac1 are denoted in Figs. 5.32 and 5.37. The detailed annotation for line types is presented in the lower-left and upper-left corner of these figures. 164
2. As known, as there is a relative sliding between piston and cylinder, it will occur the sliding friction as shown in Fig. 5.39. Herein, it is noteworthy that Fsf-c and Fsf-p are the frictional forces inserting the cylinder and piston and the denotation of “1” and “2” are representation of the cylinders 1 and 2, respectively. We have: Fsfc1 F sfp 1 F sf 1 sign( u ) (5.83) Fsf2 c F sf 2 p F sf 2 sign( uu ) in which Fsf is calculated by Eq. (5.8) From this analysis, Fig. 5.40 shows the free body diagram of the system which is subjected to the sliding friction and excited force fe=Fecos(t) with the force amplitude of Fe and frequency of  Consider virtual displacement u, the virtual works done by these forces become WF sfpsfp1 F 1  xF 1 sfpsfp 2 F 2  xMgufu 2 e (5.84) herein, x1 and x2 are the corresponding virtual displacements of the piston 1 and 2. These are the same as the virtual displacements of the center of the rollers 2 and 3 as defined in Fig. 4.9(b). Applying Eq. (4.22), (4.27) and (5.83), Eq. (5.84) can be recast, by u (5.85) W 2 Fiuusf2 s gn( ) 2 Fiu sf 1 s gn(  ) tan  Mgfu e 2 2 R r u Accordingly, the generalized force in the direction of the relative coordinate u is obtained as below: u QfFiuu esf22 s gn( ) 2 Fiu sf 1 s gn(  ) tan Mg (5.86) 2 2 R r u 166
3. u MucuF s2 Fi sf1 s gn( u  )tan 2 F sf 2 si gn( uu  ) Mg F e cos  t (5.89) R r 2 u2 1.8 1.6 1.4 1.2 1.0 u 0.8 a 0.6 0.4 0.2 0.0 -0.2 0 2 4 6 8 10 Fig. 5.41. Multi-scale method compared with numerical integration First of all, the amplitude-frequency curves obtained by Multi-scale method and fourth-order Runge-Kutta algorithm are compared as shown in Fig. 5.41. Herein, the frequency of the excited force is swept up slowly from 0 to 10 rad/s (denoted by the dashed line). In contract to reducing slowly frequency 10 rad/s to 0, the amplitude- frequency curve is exhibited by the dot line. It can be seen that although at the down jump point, there is a difference between two methods due to approximate error and exiting sliding friction between the piston and cylinder, two curves are still in good agreement. Next, the complex behavior of the proposed system will be investigated through the numerical integration for Eq. (5.89) using a fourth-order Runge-Kutta algorithm with the various initial conditions including velocity and position. The bifurcation diagram ˆ of Eq. (5.89) for Vac1 7.89, µ=1.834,  changed from 1 to 10 rad/s and the same other parameters as in Fig.5.30. The value of u is determined by using Poincare map with the period T=2 /. The simulated result is plotted in Fig. 5.42. It can be seen that if the 168
4. response will be detected in three cases following. In the 1st study case, the parameter  is taken account into at value of 2.4 rad/s, simultaneously other parameters are set as in Fig.5.42. As shown in Fig.5.42, it can be seen that it may exist the period-1 or period-2 solution depending on the initial condition. It is interesting to see that in the attraction basin depicted in Fig.5.43, the attractor region of the period-1 oscillation is greater than that of the period-2 one. This means that the ability to obtain period-1 steady dynamic response is higher. Additionally, the vibration ability of the period-1 oscillation plotted in Fig. 5.44 (b) is higher than that of period-2 one depicted in Fig.5.44 (a) in which the initial position and velocity of the first solution is zero but the u 20 mm and u 0.2 m/s, second one is obtained for o o meanwhile the fixed points calculated by Poincare section are annotated by filled circles. o  u Fig.5.43. Attractor-basin phase portrait for =2.4 rad/s, other parameters set as in Fig.5.42 170
5. occurred is narrower than that in the 1st case. Furthermore, in this case as shown in Fig.5.46 (the detailed annotation of line types is presented in right-top corner panel of each figure), the vibration level of the period-1 solution (Fig. 5.46(b)) is lower than that of period-3 one (Fig. 5.46(a)). Both solutions also revealed that the vibration of the system occurs at a position which is drifted away from the equilibrium position (u=0). 0.20 0.12 Vˆ 7.89 ˆ ac1 Vac1 7.89 Vˆ 26.30 ˆ 0.15 ac1 Vac1 26.30 0.08 0.10 0.05 0.04 0.00 0.00 (m/s) (m/s)   u -0.05 u -0.04 -0.10 -0.15 -0.08 -40 -30 -20 -10 0 10 20 30 40 -10 -5 0 5 10 15 ˆ Fig. 5.46. The phase orbits of the system for Vac1 7.89; 26.30 and uo 20 mm and u o 0.05 m/s(a); uo 0 and u o 0(b) In the 3rd study case, the dimensionless volume of the auxiliary chamber is increased to the value of 26.30 but other parameters and frequency are the same as in the 2nd case. To guarantee that the minimum stiffness is nearly zero, the pressure ratio is calculated at value of 0.997. The result is that the area of the period-1 solution is expanded compared with the second case as shown in Fig.5.47. Similar to the 2nd case, the amplitude and velocity of the period-1 solution are reduced compared with the period-3 oscillation as shown in Fig.5.46 in which the phase orbits are drawn by the dashed line meanwhile the fixed point is annotated by square. This case confirms that the position at which the load plate oscillates around was moved to the equilibrium position. 172
6. ˆ K DSEP ˆ ˆ Ks K DSEP ; uˆ 0 Fig. 5. 48. Flow chart for designing the QSAVIM using PC 174
7. asymmetry of the stiffness curve will be reduced along with growing the volume of the auxiliary chamber. Then, the analysis of equilibrium position and the procedure for designing the QSAVIM using PC having the lowest stiffness value around the DSEP investigated and suggested. The fundamental resonance response and force transmissibility of the QSAVIM using PC subjected to the externally harmonic force is analyzed through Multi-Scale method and the numerical simulations are verified. The simulation indicated that because of the effect of asymmetrical stiffness curve, the QSAVIM using PC can be a soft or hard system depending on the auxiliary chamber volume and pressure. It also confirmed that the lower the dynamic stiffness is, the larger the effective isolation region is and the better the isolation effectiveness is. Especially, the lower the asymmetry level of the stiffness curve is, the more the effectiveness of suppressing the force transmissibility from the load plate to the base is improved. Additionally, parameter bifurcation analysis of the QSAVIM using PC had been realized through numerical integration from the original dynamic equation. Simultaneously, fixed points had been also calculated by using Poincare map. The result proved that the system can occur period-1, period-2 or period-3 oscillation depending on the initial conditions. 176
8. contradiction. Especially, the proposed model can be fabricated and applied certainly in Viet Nam. The study result proved that the adjustment of the stiffness of both mechanisms is realized easily through controlling the air pressure in the air spring. In addition, the operation of the proposed model can be easily transferred from passive into active state to obtain the wanted isolation response. Whilst, it is very difficult for the quasi-zero stiffness vibration isolator using mechanical springs to realize this mission. Specifically, the result of this study obtained as following: 1. A QZS vibration isolation model using rubber air springs The physical parameters such as effective area and volume of a commercial rubber air spring were built and identified experimental. Then, the restoring force model as well as stiffness of the air spring due to compressed air was obtained. Moreover, because of inheritance of the rubber material which includes the friction between reinforce fiber and rubber, and viscoelasticity, the hysteresis curve of the rubber air spring was also identified experimentally through Berg’s model and fractional Kelvin- Voigt’s model. The result confirmed that model of the rubber air spring contributed by compressed air, friction, and viscoelasticity follows well the experimental data. Based on the result obtained from rubber air spring model, the stiffness equation of the QSAVIM was established. Then numerical simulation of the stiffness curve was realized meaning that the stiffness curves is a symmetrical concave parabola around the DSEP. Over expected working range, the dynamic stiffness of the QSAVIM is lower than that of the ETVIM. The pressure ratio, that is the pressure ratio of the load bearing mechanism to the stiffness corrected one, is obtained, indicating that the dynamic stiffness of the QSAVIM is increased according to the growth of the pressure ratio. Thank to this relation, the pressure of both mechanisms can be easily adjusted so that the quasi-zero dynamics stiffness of the proposed system is always remained at the 178
9. the ETVIM. Indeed, the obtained result is that the proposed model can presented the attenuation of vibration transmission from the source to the isolated object in frequency region larger than 31.5 rad/s (5Hz) 2. A QZS vibration isolation model using pneumatic cylinders To show comprehensively the quasi-zero stiffness vibration isolation model using air springs, in this thesis, the pneumatic cylinders connecting auxiliary tanks were also considered as elastic elements. First of all, the stiffness model of the pneumatic cylinder was obtained by the analysis solution based on thermodynamic equation and ideal gas. The sliding friction between the piston and cylinder was then taken into account. Instead of experiment, this thesis utilized the development of software technology and virtual prototyping technique. Particularly, a virtual model of the pneumatic cylinder adding an auxiliary tank was built to evaluate the analysis model of the pneumatic cylinder and identify the sliding frictional model. The result of the virtual simulation confirmed the accepted accuracy of the analysis model. Next, the stiffness model of the load bearing mechanism (LBM) using the pneumatic cylinder adding the auxiliary tank was drawn and analyzed. The simulation result shown clearly that the stiffness of this mechanism is not a constant value that it will be changed with respect to the position of the load plate. Moreover, it is a nonlinear and asymmetric curve around the DSEP. The asymmetry and nonlinearity will be reduced in accordance with the increase in the volume of the cylinder adding auxiliary tank. This means that when this volume is large enough, the slope of the stiffness curve is very small. Then, stiffness model of the stiffness corrected mechanism (SCM) using the pneumatic cylinders adding auxiliary tanks was also obtained. Unlike stiffness curve form of the LBM, the stiffness curve of the SCM is always a symmetric parabola round the DSEP. This parabola can be concave or convex depending on the volume of the cylinder connecting tank. The analysis result indicated that when the auxiliary volume is increased from zero to a critical value, the stiffness 180
10. - Studying the damping methods to reduce peak frequency - Studying control algorithms to improve the isolation performance Published papers International Journal 1. N.Y.P Vo and T.D. Le, “Adaptive pneumatic vibration isolation platform”, Mechanical Systems and Signal processing, 133,106258, 2019. (ISI, Q1, IF=6.832, H-index=167). 2. Ngoc Yen Phuong Vo and Thanh Danh Le, “Static analysis of low frequency Isolation model using pneumatic cylinder with auxiliary chamber,” International Journal of Precision Engineering and Manufacturing, 21, pp. 681- 697, 2020. (ISI, Q2, IF=2.106, H-index=50). DOI: 10.1007/s12541-019-00301-y 3. N. Y. P. Vo, T. D. Le, “Analytical study of a pneumatic vibration isolation platform featuring adjustable stiffness”, Journal of Commun Nonlinear Sci Numer Simulat, 98, 105775, 2021 (ISI, Q1, IF=4.26, H-index=113). 4. Ngoc Yen Phuong Vo and Thanh Danh Le, “Dynamic analysis of quasi-zero stiffness pneumatic vibration isolator”, Applied Science, 12, 2378, 2022, (ISI, Q2, IF=2.679, H-index=52). 5. N.Y.P Vo, M.K. Nguyen and T.D. Le, “Dynamic Stiffness Analysis of a Nonlinear Vibration Isolation Model with Asymmetrical and Quasi-Zero Stiffness Characteristics”, Journal of Polimesin, 19, pp. 7-15, 2021. (IF=0.65). 182
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