华侨大学陈振川全球求助翻译英语论文_(12).pdf

83 The Fifth International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY Huazhong University of Science and Technology Wuhan P R CHINA 27 28 August 2009 2009 Huazhong Universiti of Science and Technology Press ACTIVE CONTROL OF FLEXIBLE VIBRATION SYSTEMS WITH INCLINED COMBINED MOUNTS Niu Junchuan a b Lim C W c Zhu Jiqing a b a School of Mechanical Engineering Shandong University Jinan P R China b Key Laboratory of High Efficiency and Clean Mechanical Manufacture Shandong University Ministry of Education P R China c Department of Building and Construction City university of Hong Kong Hong Kong P R China Email niujc ABSTRACT In order to reduce the vibration and noise of flexible systems subjected to multi dimensional excitations efficiently the paper presents a general model of isolation systems with passive active inclined mounts by subsystem mobility technique and transfer matrix approach In the model the passive mounts are fixed to the seats to react against axial force transverse force and couples and inside actuators which are parallel with the passive mount and pined pined at the seats produce axial active forces The power flow transmitted into the foundation is considered as a cost function for optimal control and investigating the transfer characteristics of vibration Some numerical examples are presented to show the effectiveness and the feasibility of the model and also some useful conclusions are obtained KEYWORDS active control flexible system inclined mount I INTRODUCTION In mechanical engineering some vibrating machines such as engines and compressors need to be supported by various mounts to isolate imposed vibration and keep silence In order to achieve excellent isolation performance numerous researches on the passive and active mounts have been investigated As a kind of widely used passive mounts rubber isolators which have low damping show efficient vibration isolation performance in the non resonant and high frequency excitation 1 2 The traditional installation of the rubber mounts is always that the principle elastic axes of the rubber isolators run parallel to the principle inertia axis of the supported machine In these cases the dominated vibration is along vertical inertia axis of the machine when only a vertical force is applied to the center of gravity of the machine and the mounts are subject to axial compression loadings 3 But in other cases of the complex excitations which are combined by vertical and horizontal forces and couples the vertically installed mounts can not isolate the corresponding vibration efficiently even result in the failure of isolation Otherwise inclined mounts can react against the compression bending even buckling loadings which is especially meaningful for the vibration isolation of machines subjected to combined excitations Though the advantages of the inclined mounts are found earlier they are only discussed in condition of rigid foundations 3 As far as flexible bases which mounts installed on are concerned inclined mounts do not obtain enough attention until now 4 5 Moreover due to more strict requirements of isolation of some high speed and high precision instruments which are installed on the flexible 84 foundations increasing numbers of novel active control techniques are introduced to depress the vibration and noise 6 9 and once inclined passive mounts and parallel active mounts or actuators are combined to install on the flexible foundations the characteristics of the isolation systems must be explored and discussed for achieving substantial vibration attenuation In order to reduce the vibration and noise of flexible systems subjected to multi dimensional excitation efficiently the paper presents a novel model of the flexible isolation system with inclined passive active mounts The hollow rubber cylinders and parallel actuators are combined to support the rigid machine and isolate the vibration transmitted into the foundation The subsystem mobility technique and transfer matrix approach are employed to formulate the power flow The power flow transmitted into the foundation is considered as cost function for optimal vibration control At last some simulations are performed to investigate the transfer characteristics of vibration and show the effectiveness of the presented model II DYNAMIC ANALYSIS OF ISOLATION SYSTEM In order to simulate the isolation system in practical engineering the dynamic model of the system is given as shown in Fig 1 a In the model the foundation is considered as a rectangular flexible plate simply supported at the four edges The machine is rigid and supported by two inclined combined mounts The passive active mounts are composed of passive isolators and inside actuators as shown in Fig 1 b The passive mounts are generally hollow rubber cylinders which are fixed to the rigid seats so as to react against axial and transverse forces and couples and the active mounts can be hydraulic servo or other electromagnetic actuators which are pined pined at the joints with the seats and produce axial active forces d omo Jo Moy1b o1bMoy2 o2Fox1b u o1bFox2b u o2b Foz1b w o1bFoz2b w o2b Moy o Fox u o Foz w o Frx1 u r1 Frx2 u r2 Frz1 w z1 Frz2 w r2 Mry2 r2 a1a2 b1 x2 x1 Fmz1t w m1t Fmx1t u m1t Mmy1t m1t Fmz1b w m1b Fmx1b u m1b Mmy1b m1b 1 Fmz2t w m2t Fmx2t u m2t Mmy2t m2t Fmz2b w m2b Fmx2b u m2b Mmy2b m2b 2 Mry1 r1 b2 a b Fig 1 a Dynamics and Kinematics of the isolation system with inclined mounts b the installation of the passive active mount combined by passive fixed fixed isolator and pined pined active actuator 85 2 1 Dynamics Analysis of Subsystems The machine s dynamic characteristics were studied using mobility technique According to the dynamics of the rigid body the governing equation of the machine is given by 1112 2122 oo obob VFAA VFAA 1 where TT ooxozoyooxozoy FFFMVVV are the general multi dimensional force acting on the center of gravity of the machine and corresponding velocity and also obob FV are the general force acting on the bottom of the machine and the corresponding velocity respectively and kl A 1 2 k l are the sub matrices of the mobility of the machine which can be formulated using the method demonstrated in literature 9 For the passive rubber cylinders neglecting the masses of the isolators the complex stiffness matrix of the mount system is given by 12 diag diag iixizi KK KKkkk 1 2 i 2 where 1j isisis kk 1 2 isx z is the complex stiffness is is the loss factor of damping of the passive mounts When the active forces of the actuators are included in one can obtain the governing equation of the mounts as j mtmtmb FK VVQU 3 where T 12 QQ Q is weight matrix of the control output vector for pined pined actuators T 1 0 1 0 0 0 0 Q T 2 0 0 0 0 1 0 Q and T 11122211 mtmx tmz tmy tmx tmz tmy tmtm tm t FFFMFFMVuw T 1222 m tm tm tm t uw are the generald force and the corresponding velocity of the mounts and U is the control output vectors of the two actuators In the paper the dynamics of the thin rectangular plate with simply supported edges can be described as rr VCF 4 where T 111222 rrxrzryrxrzry FFFMFFM and T 111222 rrrrrrr Vuwuw are the general force subject to the foundation and the corresponding velocity respectively and C is a 6 6 dimensional mobility matrix of the thin rectangular plate with simply supported edges which can be referred to the text 10 2 2 Transfer Matrices of Dynamics and Kinematics Because of the inclined mounts being used in the isolation system the transfer relationships of dynamics and kinematics at the mounts ends must be determined for the combination of the subsystem in the next section According the showing of Fig 1 a the relationship of the general forces acting on the machine s bottom and the top of the mounts is given by obftmt FT F 5 where the transfer matrix of dynamics at the joints of the top of the mounts is given by 1122 1122 21212222 cossin0cossin0 diagsincos0 sincos0 cossin1cossin1 ft T bbbb 6 where 12 are the inclining angle of the mounts 1 and 2 with respect to the z axis And the relationship of the kinematics at the top end of the mounts is given by mtvoob VT V 7 where the velocity transfer matrix at the joints of the top of the mounts is determined by 86 11212222 11212222 cossincoscossincos diagsincossin sincossin 001001 vo T bb bb 8 Similarly for the relationship of the forces at the junction of foundation and the bottom of the mounts one has rfbmb FT F 9 where the force transfer matrix at the corresponding joints is 1122 1122 1122 cossin0cossin0 diagsincos0 sincos0 cossin1cossin1 fb T dddd 10 and the relationship of the kinematics of the foundation and the mounts is given by mbvrr VT V 11 where the velocity transfer matrix at the joints of the bottom of the mount and the foundation is 111222 111222 cossincoscossincos diagsincossin sincossin 001001 vr dd Tdd 12 2 3 Combination of Isolation System In the system the masses of the mounts is so small in comparison with the machine and the foundation that they can be omitted in the formulation thus the forces acting on the bottom and the top of the mounts is equal that is mtmb FF 13 combining Eqs 1 3 4 5 7 9 11 and 13 one can obtain the general force and it corresponding velocity transferred to the foundation as 12ro FS FS U 14 34ro VS FS U 15 where 121 vofb ST TKAT 2 fb TSTQ 42 SCS and 1 22 voftvrfb TIK T A TT CT The equations above are based on the general vibration system If the active terms in Eqs 14 and 15 are set to zeros the active isolation system will degenerate into a passive one III CONTROL STRATEGY OF POWE FLOW In controlling the vibration of coupled flexible systems both forces and velocities transmitted through mounts into the foundation should be considered Therefore power flow is considered as an index for achieving optimal control of the vibration transmission into the foundation In a vibrating system the power flow transmitted into the foundation is given by 11 HHHHH 1 4 rrrrr PFVVFU AUU BB UD 16 where HH 2442 4AS SS SQ HH 2341 4 o BS SS SF HH 1331 4 H oo DFS SS SF and superscript H indicates the complex conjugate and transpose of matrices or vectors It is observed that Eq 16 is of Hermitian quadratic format Its unique minimum is assured provided that the matrix A is positive definite The corresponding optimal voltage vector yields 12 1 opt UA B 17 Substituting the optimal vector shown in Eq 17 into Eq 16 the controlled optimal power flow 87 transmitted into the foundation can be computed easily IV NUMERICAL EXAMPLES To demonstrate the effectiveness of the presented model some numerical examples are presented here In the examples some parameters are listed below mo 293 7kg Jo 8 4kg m2 k1x k2x 3 6e5N m k1z k2z 7 8e5N m k1 k2 1 2e3N m 1 2 x1 0 535m x2 1 115m and the length width height density elastic modulus poison ratio of the rectangular plate are a 1 5m b 1 0m h 0 01m 2700kg m3 and 0 33 respectively Fig 2 shows the power flow transmitted into the foundation in case of different inclining angles and only horizontal excitation force being applied In the figure the first two peaks are respectively the transverse mount model along the x axis and the angular mount mode around the y axis and the translation mode along z axis is not activated in this case With the increase of the inclining angles the first two peaks are becoming more and more close and the power flows transmitted into the plate are increasing in the mid and high frequency range when no active control is included in For active control it is observed that the effectiveness of the control is remarkable for reducing the vibration with the increasing of the inclining angle of the mounts especially in the high frequency domain Fig 3 gives the influences of the different inclining angles of the mounts on the controlled and uncontrolled power flows when only vertical force is applied In this case only the vertical mount mode is aroused The peaks of the power flow are varying with the variety of the inclining angle slightly The active control reduces the power flow significantly and with the increase of the inclining angle from 0 to 90 degree the effectiveness of the actuators is becoming weaker In the figure the plot of controlled power flow in case of the inclining angle being zero is not presented because the foundation under control keeps almost static condition so that the power flow is below 10dB and unable to be printed in the presented figure which has been reported in the literature 13 1101001000 10 5 0 5 10 15 20 Power Flow P dB Frequency f Hz 0 Uncontrolled 0 Controlled 30 Uncontrolled 30 Controlled 60 Uncontrolled 60 Controlled Fig 2 Power flow into the foundation in case of excitation Fo 1 0 0 Fig 3 Power flow into the foundation in case of excitation Fo 0 1 0 1101001000 10 5 0 5 10 15 20 Power Flow P dB Frequency f Hz 0 Uncontrolled 30 Uncontrolled 30 Controlled 60 Uncontrolled 60 Controlled 90 Uncontrolled 90 Controlled Fig 5 Power flow into the foundation in case of excitation Fo 1 1 1 1101001000 5 0 5 10 15 20 Power Flow P dB Frequency f Hz 0 Uncontrolled 0 Controlled 30 Uncontrolled 30 Controlled 60 Uncontrolled 60 Controlled Fig 4 Power flow into the foundation in case of excitation Fo 0 0 1 1101001000 5 0 5 10 15 20 Power Flow P dB Frequency f Hz 0 Uncontrolled 0 Controlled 30 Uncontrolled 30 Controlled 60 Uncontrolled 60 Controlled 88 Fig 4 illustrates the power flow transmitted into the foundation in case of only a couple being considered In this case the couple arouses the vertical rigid mount mode and the angular mount mode It is obvious that increase of the inclining angle gives good isolation performance for the passive system However although the actuators reduce the total power flow obviously the effectiveness of the active control is becoming weaker with the increase of the inclining angles of the mounts Fig 5 demonstrates the power flow transmitted into the foundation when a three dimensional excitation acts on the machine It illustrates similar results as shown in Fig 4 except that the three mount modes appear So the discussion is not repeated V CONCLUSIONS The paper presents a model of isolation system with passive active inclined mounts by subsystem mobility technique and transfer matrix approach The power flow transmitted into the foundation is considered as the cost function for optimal vibration control Some numerical examples are presented to show the effectiveness and the feasibility of the model and also some useful conclusions are obtained The inclining angle has influences on the power flow transmitted into the flexible foundation With the increase of the inclining angle the uncontrolled power flow is shifting slightly and the effectiveness of the actuators on power flow is becoming little weaker when only vertical force or coupl