Tài liệu 7.mimmi1999 luc moment

G. C. Mimmi Universite degli studi di Pavia Dipartimento di Meccanica Strutturale Via Ferrata, 1,1-27100 Pavia, Italy E: [email protected] P. E. Pennacchi Politecnico di IVIilano Dipartimento di IVleccanIca P.zza L. da Vinci, 32,1-20133 Miiano, Italy E: [email protected] Dynamic Loads in the ThreeLobe Supercharger In previous papers the authors have analyzed the functioning of the three lobe supercharger. This analysis was made possible by a complete analytical model of the rotors. The dynamic transformation of the chambers occupied by the fluid allows us to point out specific problems regarding trapped volumes and to design special devices to avoid these problems. As a result it is possible to simulate the performance of the compressor and to propose an analytical model in order to determine the dynamic loads on the rotors. The topic of this paper is the comparison between the performances achieved by helical and spur rotors, with particular attention to the loads, force and moment, acting on the rotors. Introduction The scientific interest in positive displacement blowers used as superchargers of internal combustion engines has been increasing recentiy. The reason for this is that the use of superchargers permits us to increase torque at low speed and to reduce emissions (Middlemiss and Noble, 1993). Many references exist in literature relating to different types of superchargers such as screw compressors (Curtis, 1991; Matsubara et al., 1989), rotary piston superchargers (Heikrodt, 1989) or combination of turisochargers and mechanical chargers (Schmitz et al., 1994). Among the types proposed, the Roots type has found practical application in the automotive field. The most common type of this kind of compressor (also for other applications) uses two lobe spur rotors (Stone, 1988; Kestin and Owczarek, 1952). Nevertheless some cases of three lobe spur rotors are known. Three lobe Roots compressors of helical shape have appeared recently. Their lower noise emission in respect to that of the corresponding spur rotor type is emphasized in literature (Huttebrauker et al. 1995 and 1996). Moreover they give a better motor performance and fuel consumption and emission are reduced. (i) (ii) inlet and outlet pressure values remain constant; there are perfect seals between the elements of the blower and no leakage. The method usually adopted to determine the dynamic loads represents them as a force and a moment that are calculated by integration, starting from their value on an infinitesimal surface (Adams and Soedel, 1995; Mimmi and Pennacchi, 1998b). Let di, be an infinitesimal surface element, in the neighborhood of a general point Q, which belongs to a surface 2(p, u), on which the pressure p acts. The general equations of the infinitesimal components of force and moment around the midpoint of each rotor rotation axis, for the considered surface element, are: rfF = pndl Calculation of Pressure Dynamic Loads An analytical model of the rotors typical of these blowers is proposed in a previous paper by the same authors (Mimmi and Pennacchi, 1998a), while a mathematical model for calculating the dynamic loads due to the fluid is reported in another paper by the same authors (Mimmi and Pennacchi, 1998b). The last model is rather complex: starting from a 3D analytical model of the rotor, the theoretical contact lines between the rotors are determined (see Fig. 1). These allow us to determine the exact shape of the chambers occupied by the fluid. An accurate analysis of the chambers which convey the fluid during the rotation of the rotors is presented by means of a study on an axial section. An example of the chambers formed in a particular angular position of the rotors is presented in Fig. 2. One may observe that, in the hypothesis of perfect seals, some of the chambers may close and later may shrink without opening toward the exhaust. For this reason special auxiliary grooves are introduced (Mimmi and Pennacchi, 1988b) and designed in detail. By means of this procedure it will be possible to evaluate the loads due to the fluid under pressure, which will be modeled by a force and a moment and calculated by integrating the pressure of the fluid on the surface of the chamber belonging to the rotor. Contributed by tlie Design Automation Conimittee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received Nov. 1998; revised Aug. 1999. Associate Technical Editor; H. Lanltarani. 602 / Vol. 121, DECEMBER 1999 These last results are presented in detail herein and only a short description is given to illustrate the calculation of the loads. Moreover the results obtained for spur and helical rotors are compared and the differences in load are stressed. In order to calculate the dynamic loads due to the pressure, the following hypotheses were taken into consideration: r Xr = p ,, " ^ ^"jj- llfp X VMP du • piVp X r j d p du dM = (Q- 0)XdF = (Q- (1) 0)X pitp X r„)dp du (2) The procedure is rather complex, especially if we consider that all the integration regions have to be determined, and their boundary should be defined as functions of the surface parameters employed for generating the 3D analytical model and of the rotor rotation angle. Moreover, the value of the pressure p is a function of the angular position of the rotors. The details of the calculation are reported in (Mimmi and Pennacchi, 1998b). It is clear that the two rotors have to be considered separately for the calculation, however, a symmetry relationship exists which allows us to obtain the components of the force R2(^) on the second rotor once the force Ri(#) on the first are given: r«2,(d)=-i?„(#+T7/3) R2W = RiyW [R,,m = Riyi-& + 'rr/3) = R,,i^+77/3) (3) A similar relationship also exists for the resultants S,{d) and S2(^) of the moments: rs2,(#) = s „ ( # + T7/3) S2i&) = \S2yW=-Sy{'&+7T/3) (4) [S2,(#)= - 5 „ ( # + 7 r / 3 ) Copyright © 1999 by ASME Transactions of tiie ASME Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jmdedb/27665/ on 02/20/2017 Terms of Use: http://www.asme.org/ mMitili-i'l-ri I i Fig. 1 Contact line between the rotors Fig. 2 Chamber determination on axial section Analysis of the Calculated Loads plates, which correspond to the rotation angles {^ = 4>, •& = T7/3 — (/),#= 7r/3 + (/) and # = 27r/3 - (/>. The grooves cause a sudden pressure variation from p , to p2 in chambers that have a large total surface, since they have an axial length equal to the rotors, though the relative regions have limited Resultants of Forces for Helical Rotors. The components of areas on the axial section. resultant R, of forces of the first rotor are shown in Fig. 3 as a Even though this phenomenon was also present for 7 = 60 deg, function of the rotor rotation angle •& in the range from 0 to ITTB, it was not evident in Fig. 3, since the chambers connected to the and for the following design parameter values: r,, = 21.8 mm, r, = 32.2 mm, A = 3 r „ p , = 0.1'MPa,p2 = 0.2 MPa, y = 60°. grooves, as a consequence of the helical shape, have an axial These are sample values and similar results to those described length less than h. Component R\y: the same observations made for i? J, also apply from now on can be obtained by using other values of the design to the component R\y with the exception that it is always positive parameters. in the period. Component /?IJ:: by analyzing the diagram of component R i^, a Component Rl/, the component R], has a null value since the considerable jump effect (about 400 N) can be observed. This rotor does not have a helical shape. jump is due to the pressure rise in the chamber as soon as it is connected to the outlet, for a rotation angle equal to •& = TT/6 + (p. Comparison Between the Resultants of the Forces. It is Only the component /?,, changes its sign. interesting to analyze the resultant components of the forces for Component /fi,: the component R,y has a high average value both 7 = 60 deg (helical) and 7 = 0 deg (spur) along the same during the rotation with low superimposed amplitude harmonic axis. In particular. Fig. 5 shows the comparison of the two types component. In this case the opening effect of the chamber, for a along the x axis. rotation angle equal to i9^ = 7T/6 + (j>, is not relevant. It is possible to observe that both components have a similar Component /Jj^: the component /? i^ along the rotation axis has amplitude, but a greater number of jumps is present for 7 = 0 deg. a limited value during the rotation period, with reduced oscillations Moreover the average value of R1, is lower for 7 = 60 deg. at about the average value. These are lower than those of the radial The comparison between the components along the y axis (Fig. components R,, and R^^. 6) shows that the spur rotors present a higher number of harmonic components and the amplitude of the load is much greater than that Resultant of Forces for Spur Rotors. The components of on the helical ones. resultant R\ of forces of the first rotor are shown in Fig. 4 as Nevertheless, the helical shape causes an axial load (R i^ in Fig. functions of the rotor rotation angle •& in the range from 0 to 2T7-/3, 3), which is however reduced and without jumps, while the axial and for the same design parameters of the previous case, with the load in the spur rotor is zero. exception of 7 = 0 deg. Resultant of Moments for Helical Rotors. The components Component/JJj: the component R'u shows two jumps. The first corresponds to the opening of a new chamber towards the outlet, of the resultant S1 of moments of the first rotor with a helical shape which is delayed in this case to the rotation angle •& = Tr/2 + 4> are shown in Fig. 7 as a function of the rotor rotation angle # in because the rotors do not have a helical shape. The second jump is the range from 0 to 2'7r/3. relative to the opening or closing of the grooves on the lateral Component Si,: the component presents an evident jump due to We present here an overview of the results of a simulation obtained by calculating the loads both for helical and spur rotors for a particular set of design parameters. Nomenclature F = force; M = moment; n = vector normal to surface 2 ; p 1 = inlet pressure; P2= outlet pressure; Ki, = base radius; Journal of Mechanical Design r, = r, = r,, = r = R, = tip radius; inner radius; pitch circle; surface vector; resultant force on i" rotor; S, u 7 d p = = = = = resultant moment on i* rotor; surface parameter; helix angle; rotor rotation angle; surface parameter; DECEMBER 1999, Vol. 121 / 603 Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jmdedb/27665/ on 02/20/2017 Terms of Use: http://www.asme.org/ i?[N] > 400 200 ^\x a[r •^Iz 0.5 1 T^. ^ ^ ^ 2^ -200 Fig. 3 1 Forces on the helical rotor 2 Fig. 6 the cause described previously and related to the opening of the chamber towards the outlet. The component changes its sign as with the force component in this case too. The amplitude of the oscillation is less than 3.2 Nm. Component Si^,: the component 5|j oscillates greatly with an amplitude equal to about 2.4 Nm and an asymmetrical behavior during the rotation. Component Sj^: the axial component 5,2 has a sinusoidal-like behavior with an amphtude equal to about 0.48 Nm. Resultant of Moments for Spur Rotors. The components of the resultant SJ of moments of the first rotor are shown in Fig. 8 as a function of the rotor rotation angle # in the range from 0 to 27r/3. Component SJj.: the component 5^ always has a null value due to the choice of the pole O for the moment calculation. Component S',,: the component S\y has null value for the same reason as does SJ,. Component S\^, the axial component S\, presents a remarkable 3 4 5 Comparison of forces along the y axis quasi-impulsive behavior, with an amplitude of about 4.4 Nm, due to the groove actions in correspondence with the values # = t^, # = T7/3 - ( / ) , - & = Tr/3 + f^ and # = 2^/3 - <^. The same impulsive loads are not present in correspondence with the opening of the chamber relative to the region A 3 since the acting loads in this case are balanced. The component S\, is the moment opposite that of the rotor rotation and is consequently particularly meaningful. Comparison Between the Resultants of the Moments. The comparison of the components of the moment along the z axis for both 7 = 0 deg and 7 = 60 deg is shown in Fig. 9. The remarkable reduction of the impulsive loads determined by the helical shape is evident from this diagram. Frequency Analysis of the Loads. The diagrams of the discrete Fourier spectrum analysis of force and moment components are shown in Figs. 10-12 for both 7 = 0 deg and 7 = 60 deg. ^[Nm] Fig. 5 Comparison of forces along the x axis 604 / Vol. 121, DECEMBER 1999 Fig. 8 Moments on the spur rotor Transactions of the ASME Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jmdedb/27665/ on 02/20/2017 Terms of Use: http://www.asme.org/ 5[Nm] ^[Nm] d[rad] ^L -0.8 0.96 0.80 0.64 -1.6 0.48 -2.4 0.32 -3.2 0.16 A \/A V V -4.0 5 Fig. 12 Fig. 9 ••¥l¥Wt>Mfcl'il'i 10 15 20 25 Harmonic components of moments along the z axis Comparison of moments along the z axis The order k of the harmonic component is reported on the abscissa axis while the amplitude of the k''" harmonic is reported on the ordinate. By considering Fig. 10, we may to observe once again that the highest amplitudes of the harmonic components along the x axis are comparable. As regards Fig. 11, the amplitude of the component along the y axis of the helical rotors is more reduced than in the case of the spur rotors. The comparison of the harmonic components of S,^ and S',^ is shown in Fig. 12. Once again it is possible to observe the amplitude differences between the helical and the spur rotors, that confirm the previous considerations regarding the reduction of the load amplitudes achieved by using the helical shape. Conclusions This paper presents the results of the calculation of the dynamical loads due to fluid actions on the rotors of a Roots type blower with three lobes both for helical and spur rotors. In the simulated cases the calculations of the loads, force and moment, have shown that the helical rotor shape has a positive effect and produces a lower variation in the resulting loads. In fact their amplitude is reduced in comparison with that of spur rotors for a fixed pressure variation. However, the helical shape introduces new load components along axes which do not exist in the case of spur rotors. The reduction of the amplitude of the loads is important in order to obtain a reduction of the vibration and noise of the system, as reported by experimental studies in literature. Acknowledgments The authors wish to thank M.U.R.S.T. (Ministero deH'Universita e della Ricerca Scientifica e Tecnologica) and the Universita di Pavia for funding the research. They also wish to thank Mr. Oslo and Mr. Venegoni for their helpful collaboration. References Fig. 10 Harmonic components of forces along the x axis R[N] 400^ 320 R ly 240 160 80 • —WMMM 5 Fig. 11 10 15 20 Harmonic components of forces along the / axis Journal of Mechanical Design 2? Adams, G. P., and Soedel, W., 1995, "Computation of Compression Loads in Twin Screw Compressors," ASME JOURNAL OF MECHANICAL DESIGN, Vol. 117, pp. 512-519, Curtis, A., 1991, "Is Blowing in the Wind?," Car Design Technology, Nov. 91, pp. 46-50. Heikrodt, K., 1989, "Rotary Piston Supercharger for Spark Ignition and Diesel Engines," SAE SP-780, Power Boost: Light, Medium and Heavy Duty Engines, Paper 890456, pp. 43-57. Huttebraucker, D., Puchas, C, Pick, W., and Joos, K., 1995, "Entwicklungskonzept des Mercedes-Benz-Vierzylinder-Ottomotors mit mechanischer Aufladung fiir die C-Klasse," MTZ, No. 56, Dec. 95, pp. 772-775. Huttebraucker, D., Puchas, C , Pick, W., and Joos, K., 1996, "Development of the Mercedes-Benz 4-Cylinder Gasoline Engine with Compressor," VDl-Intemationales Wiener Motorensymposium, 25-26 April 1996, pp. 69-95. Kestin, J., and Owczarek, J, A., 1952, "The Expression of Work in a Roots Blower," Proc Instn Mech Engrs, 152-3, IB, 91-94. Matsubara, H., Miyashita, K., Iguchi, Y., Tanaka, S., Akiyama, K., and Nakamura, F„ 1989, "Superior Charging Technology by Screw Supercharger and High Technology Turbocharger for Automotive Use," SAE SP-7S0, Power Boost: Light, Medium and Heavy Duty Engines, Paper 890455, pp. 31-42. Middlemiss, I. D., and Noble, A., 1993, "Comparing the Performance of Turbocharged and Supercharged Engines," Autotech 93 Seminar 34, Emissions—Update and Development (2), 16-19 Nov. 93. Mimmi, G., and Pennacchi, P., 1998a, "Analytical Model of a Particular Type of Positive Displacement Blower," to be printed in Journal of Mechanical Engineering Science—Part C. Mimmi, G., and Pennacchi, P., 1998b, "Computadon of Pressure Loads in the Threelobe Supercharger," Proc. of the 1998 ASME Design Engineering Technical Conferences and Computer in Engineering Conference, Sept. 13-16, 1998, Adanta, Georgia. Schmitz, T„ HoUoh, K. D., and Jurgens, R,, 1994, "Potentiate einer Mechanischen Zusatzaufladung filr Nutzfahrzeugmotoren," MTZ, No, 55, May 94, pp. 309-315, Stone, R, C , 1988, "The Efficiency of Roots Compressors and Compressors with Fixed Internal Compression," Proc Instn Mech Engrs, Vol. 202, No. A3, pp. 199205, DECEMBER 1999, Vol. 121 / 605 Downloaded From: http://mechanicaldesign.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jmdedb/27665/ on 02/20/2017 Terms of Use: http://www.asme.org/