ACTIVE DAMPER SYSTEM DESING AND CONTROL PART A

Transcription

1 ACTIVE DAMPER SYSTEM DESING AND CONTROL PART A Rafael Luís Texera * rafael@mecanca.ufu.br Francsco Paulo Lépore Neto * fplepore@mecanca.ufu.br José Francsco Rbero * jrbero@mecanca.ufu.br * Federal Unversty of Uberlânda College of Mechancal Engneerng - Campus Santa Mônca , Uberlânda-MG Abstract. Ths paper presents a sem-actve damper desgn. A flexble metallc bellos that pumps the orkng flud through a controlled valve and produces a reactve dampng force consttutes the damper. A pezoelectrc actuator controls the valve orfce. From the bellos and the actuator specfcaton, the model of the valve s developed by the fnte element method, consderng the nteracton beteen the flud and the structure of the damper. Numerc smulatons are carred out to determne the pressure and velocty felds as functon of dfferent values of the valve closure and dfferent values of the flud velocty. These results are used to calculate the net forces actng on the model surfaces as functon of the valve closure and the flud velocty. An adaptve neural-fuzzy nference system s appled to the obtaned numercal data to generate the non-lnear fuzzy models of the valve flud dynamcs. The obtaned results sho that ths fuzzy valve model can produce a de range of the dampng force and can be used as a smulator of the proposed sem-actve vbraton damper. Keyords. Actve damper, fnte elemen, neural-fuzzy.. Introducton In several ndustral applcatons t s necessary to absorb structural vbraton energy, manly hen mpulsve forces thout any repetton pattern produce them. In these cases, the passve absorbers present lo effcency, because they are only projected for a specfc condton of operaton. Frequently, the hydraulc pston actuators are used here large dampng forces are requred. Hoever, the mplementaton, operaton and mantenance of ths type of soluton are very expensve. These actve suspensons are used by hgh performance vehcles and represent a sgnfcant amount of the car prce. The current passve shock absorbers used on regular vehcles are desgned for a nomnal behavor and don't allo dampng factor adjustments n functon of the pavement type or even for comfort or stablty operatng condtons. Ne conceptons of vbraton absorbers ere proposed, nvestgated and tested n the last years. Some desgns of actve shock absorbers and dampers are presented here. Kplng (999), Hagopan et al. (999), Glomee and Els (998) developed shock absorbers systems based on hydraulc pston hose varable orfce s controlled by an electrodynamc valve. Oh and Onoda (00) used metallc flexble bellos and electro and magneto rheologc fluds. Parker [988] and Fegel [996] nvestgated pezoelectrc actuators appled to control valves. The man proposal of ths ork s the development of sem-actve damper that uses a metallc flexble bellos, replacng the hydraulc pston used n actve suspenson of vehcles. Ths choce s attractve because metallc bellos naturally prevents flud leakage and presents lo Coulomb frcton and t s cheaper than hydraulc cylnder assembly. The bellos geometrcal confguraton can be desgned to provde the requred axal dsplacement and stffness ranges, under a nomnal nternal pressure, to mprove produced flud flo. The dampng force s generated hen the flud flos through an orfce that can be adjusted by a servo-valve. Pezoelectrc actuators, allong a larger dynamc range for the dampng control poston the valve, hen compared to an nductve electrodynamc actuator. Ths paper presents n sesson the physcal model characterstcs of the proposed damper. In sesson 3 the fnte element model of the flud flo nsde the valve s developed and the numercal smulatons of the velocty and pressure felds are carred out, as functons of the appled axal velocty and of the sze of the valve orfce. The net axal dampng forces actng on the bellos and on the valve are calculated. A neural fuzzy nference model s developed to represent the dampng forces numercal. Alternatvely, the same data set s ftted to a determnstc model for the forces, usng a nonlnear optmzaton procedure. The effcency of these to models s compared for the full doman of nput veloctes at the bellos and the avalable range of the valve orfce sze.. Characterstcs of the Sem-actve Damper The proposed sem-actve damper conssts of to man parts: a pezoelectrc actuator system controls the valve; and a hydraulc-mechancal system generates the dampng force. The hydraulc crcut operates above the atmospherc pressure and contans a volumetrc accumulator, a varable orfce, and a metallc cylndrcal bellos. The bellos s very flexble n the axal drecton and ts extremtes are connected at to ponts of the vbratng structure that experence relatve moton. The vbratng moton, appled n the bellos axal drecton, pumps the nternal flud through the valve orfce. The dsplaced flud volume travels beteen the bellos and the accumulator.

2 The flo restrcton caused by the valve yeld pressure gradents n hydraulc crcut, the dependng of the orfce sze. Ths nternal pressure appled on the bellos effectve area transversal generates the dampng force that acts on the structure. The sgnal of the flud velocty at the orfce changes opposed to de relatve velocty appled to the bellos by the vbratng structure. The system becomes actve because the valve closure can be adjusted by a postonng control system. A lnear dsplacement pezoelectrc actuator s used to mprove the frequency response and the dynamc range of the valve. The man objectve of the project s to ncorporate varable dampng to the mechancal system, assocated to each specfc poston of the valve; that s, a ne shock absorber can be set at each samplng tme of the state varables. Ths rate depends on the tme response of the actuator, on the external exctatons and on the desred dynamc performance of the vbratory system. In the dagram shon n Fgure 0, t s assumed that the flexble metallc bello s attached to a dynamc vbratory system and that all state varables are measurable by approprate sensors. Then, consderng one reference dampng coeffcent and the assocated system state, a dgtal controller ll command the pezoelectrc actuator that ll poston the flo regulator valve. Due to a control acton, the resulted orfce sze ll set the flud velocty, producng the dampng force. The bellos expanson and contracton movements produce pressure varatons nsde the hydraulc crcut. The product of the nternal pressure n the bellos by ts cross secton area results n a value of the dampng coeffcent that depends of the control acton appled to the valve. To avod structural collapse of the bellos, the closure of the valve must be lmted so that the nternal pressure s alays nferor to the maxmum alloable pressure for the bellos. Pezoelectrc Actuator Actuator Valve Flexble Metallc Bellos Optmal Dampng Force Volumetrc Accumulator Reference Dampng Dgtal Controller Dynamcs Sensors Dynamc System Fgure 0: Blocks Dagram of the proposed actve damper components The volumetrc accumulator can be a pressure vase endoed th an nternal pneumatc lung. Adjustng the pressure n the lung can set the nomnal statc pressure n the hydraulc crcut. As an alternatve soluton, a second bellos can be used as the volumetrc accumulator. The effcency of ths devce s strongly dependent on the flud vscosty and on the absence of mxed bubbles of gas dssolved n the ol. To solve ths problem a parallel crcut th a vacuum bomb can provde a prelmnary remove the gas bubbles that are present n the orkng flud. The choce of sutable value of the statc nomnal pressure n the hydraulc crcut can prevent the dssolved gas bubble generaton n the lo-pressure regon of the crcut. A commercal metallc bellos as specfed. It supports nternal pressures up to 3.3 GPa, has an effectve cross secton area of 383 mm, can generate axal loads up to 300 N th a margn of safety of 5 %. The stroke of the bellos s n range of 8 mm up to 6 mm. In ths project the valve as projected for dampng forces varyng from N to 800 N. The pezoelectrc actuator has the dsplacements range equal to 0.5 mm. It supports forces up to 570 N, as a functon of the appled voltage n the frequency band from zero to 480 Hz. 3. The valve flud dynamc model. The hydraulc crcut model s developed by the fnte element technque. As shon n Fgure, the vbratons appled at the left end of the bellos mposes a knon velocty to the flud that s n contact th ts the nternal end surface. At the rght end of the hydraulc crcut a constant pressure, equal to the nomnal statc pressure s mposed. The volumetrc accumulator s not shon n ths fgure. The valve orfce has conc shape. The orfce sze can be adjusted by movng the concal core n the axal drecton. In ths regon the flud ll reach hgher veloctes. When the bellos contracts the pressure ll ncrease at the regon located at the left sde of the core and ll be reduced at ts rght sde. If the bellos expands, the pressure dstrbuton ll be reversed. Ths model s used to determne the velocty and pressure felds developed n the flud. By ntegratng the pressure on the core surface (excludng the actuator rod) the net force actng on the valve core s determned. Applyng the same procedure on the left end surface of the bellos, the effectve dampng force actng on the vbratng structure can be calculated. The computatonal model of the hydraulc crcut as developed usng the module FLOTRAN CFD, Computatonal Flud Dynamcs, of the computatonal program ANSYS 6., consderng the K - ε model of turbulence, developed by Launder and Spaldng (974). Accordng to the geometrc characterstcs of the problem, an ax-

3 symmetrc fnte element model can be assembled. Fgure shos the confguraton of the system components and the mesh geometry constructed th the FLUID4-D quadrlateral fnte elements The mesh s refned at the central regon around the valve core, here hgher velocty gradents are expected to occur. The mesh as composed by 450 elements, snce ths confguraton provdes the numercal convergence for the velocty and pressure felds, th reasonable number of teratons Vbraton Dsplacement Flud Velocty Valve Force Pezoelectrc Actuator Dampng Force Metallc Flexble Bello Orfce Valve Obturator Fgure. Fnte element ax-symmetrc model of the hydraulc crcut. The orkng flud s a netonan lubrcant ol th densty equal to 8,87x0 kg/m3 and vscosty of 0.88 kg/ms. For all numerc smulatons the vscosty s consdered to be temperature ndependent and the densty s kept constant. The axal poston of the concal core relatve to the valve body s determned by the GAP varable measured at the X-axs of the nertal reference frame. As shon n Fgure 3, GAP s the dstance beteen the vertces of the trangles abc and ABC. The sphercal tp of the core s generated by the crcumference nscrbed n the trangle abc. The rght end of the core has a cylndrcal shape, here the actuator rod s fxed. Y C y c rao A a b B X,x GAP Fgure 3. Valve geometry and mesh refnement If the reference frames XY and xy are concdent GAP results equal to zero and the valve s totally closed. The refnement of the mesh near the concal all and the mesh geometrcal transton appled ths regon of the model s also shon. Prelmnary numercal smulatons shon that the obtaned Reynolds Number ere lo, so that the flud flo can be consdered lamnar. For steady state flo ten thousand smulatons ere accomplshed varyng combnatons of the to follong condtons: Valve closure: 50 values are set to GAP n the range of 0.8 to.3 mm, respectvely, the mnmum and the maxmum sze of the valve orfce; Vbraton relatve velocty: 50 values of axal speed Vx n the range of -0.4 to 0.4 m/s, hch physcally corresponds to the velocty resultng from the contracton or expanson of the bellos. Other fxed condtons ere adopted and mposed for problem resoluton. The statc or reference pressure nsde the hydraulc crcut s set equal to the atmospherc pressure. The appled boundary condtons are Vx= 0 and Vy= 0 at all nternal alls, except at the left surface of the bellos. Due to the symmetry Vy = 0 at the ponts here Y=0.

4 Fgure 4 sho the flud velocty and the pressure dstrbuton at the valve regon for the left face of the bellos movng at Vx=0.4 m/s. The valve core s postoned th GAP=.3 mm. Fgure 4. Velocty feld and pressure dstrbuton, for GAP=.3 mm and Vx= 0.4 m/s. For the nput velocty Vx=0.4 m/s, the maxmum velocty ampltude resulted 3.86 m/s, occurrng n the regon locate beteen the core and the valve body. At all other regons of the model the flud velocty presents much loer values. Ths ndcates that sze of the orfce, defned by the varable GAP, s domnant to determne the dampng proprety of the devce. The maxmum pressure resulted equal to 0.34 MPa and occurs at sphercal face of the core. The pressure gradent ncreases n the orfce regon and beyond. The mnmum negatve value s 0.0Mpa, meanng that t s loer than the constant statc pressure mposed n the hydraulc crcut. The pressure nsde the bellos s postve and practcally constant. Fgure 5 sho the flud velocty and the pressure dstrbuton at the central regon of the model, for nput velocty, Vx=-0.4 m/s, and the same GAP condtons of the prevous smulaton. For the case of a negatve nput velocty the flud flo s reversed, and the maxmum velocty magntude resulted 4. m/s, occurrng n the same regon. The maxmum postve pressure s 50 Pa, occurng at the rght sde of the valve core. At the bellos the pressure has negatve constant value equal to 0.37 Mpa. Ths ndcates the nverse behavour observed for postve nput velocty. Fgure 5. Velocty feld and pressure dstrbuton for GAP=.3 mm and Vx=-0.4 m/s.

5 The mnmum value of the statc pressure necessary to prevent the bubble formaton n the flud flo, can be determned usng ths smulaton, consderng that ts value must be at least equal to the pressure drop across the valve. Integratng the pressure over the core surface results the net force at the valve core. Ths force has to be supported by the pezoelectrc actuator. The dampng force appled to the vbratng structure s calculated by ntegratng the pressure over the area of the bellos left end. Fgures 6 and 7 sho the nonlnear behavor of these forces, as functons of the nput velocty, Vx, and of the valve closure, GAP. Valve Core Force [ N ] Fgure 6. Fnte elment smulatons of the forces at the valve core. Fgure 7. Fnte element smulatons of the dampng force

6 It can be observed that the forces present behavor approxmately lnear th the velocty and exponental th GAP. Ths characterstc s very nterestng snce a small change n the valve poston mplcates an expressve varaton of the dampng force. In other ords, that exponental effect reflects the hgh sensblty of the valve. Ths s necessary snce the pezoelectrc actuator, despte ts hgh force capacty, has lmted dsplacement range; tch s equal to 0.5 mm, for the selected one. On the GAP scale the value of 0.8 mm corresponds to the condton of smaller orfce, generatng larger dampng forces, and the value of.3 mm ndcates the poston of larger valve openng. For practcal operaton of the devce, an offset can be superposed to the avalable 0.5 mm range of the pezoelectrc actuator. Ths ay, the dampng force range can be set to accomplsh the actual applcaton. To physcal lmts must be taken nto account: the maxmum pressure nsde the bellos must be loer than fatgue lmt of the materal; and the maxmum force at the valve core must be loer the blocked force capacty of the pezoelectrc actuator. 4. Reduced models for the dampng and valve core forces In order to apply the controlled dampng devce to a mechancal system a reduced model for the dampng force and for the force at the actuator must be developed. The drect use of the fnte element model sn t practcal snce the computatonal tme nvolved n the numercal smulaton s to hgh for control applcatons. A frst data reducton method s to ft an analytcal model for the force functons F= f (Vx, GAP). Consderng the behavor observed n Fgures 6 and 7, a reasonable model that uses an exponental behavor th the GAP and a lnear one th Vx, s defned n Equaton. bgap ( Vx GAP) a Vx e f, = () The values of the parameters a and b are determned by a multdmensonal unconstraned nonlnear mnmzaton procedure that uses Nelder-Mead smplex method. For all GAP values, ths approach fttng errors are less than 0.% for Vx >0.00 m/s, but about 8 % for small values of the nput velocty. The model for force at the valve core has a c =-6.6 and b c = -3003, and the dampng force parameters are a d =95.4 and b d =998. The analytcal nverse model s easlly obtaned, and the GAP or Vx can be estmated from the force value. Ths ablty can be helpfull on the operaton of the controlled system. Fgures 8 and 9 present the forces estmated by the analytcal models. The GAP scale vary from 0.5 mm to 0.5 mm correspond respectvely to mm and mm valve postons. Fgure 8. Analtcal model for the Force at the valve core..

7 GAP [ mm ] Fgure 9. Analtcal model for Dampng Forces. A second data reducton method s acheved by usng a neural-fuzzy mappng of the force surfaces generated by the smulatons of the fnte element model. Ths approach leads to a non determnstc model that s determned by the method ANFIS Adaptve neural fuzzy nference system, proposed by Jang et al. (997). The unrestraned role Sugeno fuzzy model s a unversal estmator to any nonlnear functon; tch can be proved mathematcally by the Stone- Weerstrass theorem. A to nputs (x, y) and one output (z) Sugeno frst order fuzzy system ll be used to understand the ANFIS archtecture, shon n Fgure 0. For sake of smplcty, lets assume the follong rules for the Sugeno system: Rule : f x s A then f = p x + q y + r Rule : f y s A then f = p x + q y + r A B f = p x + q y + r z =.f + +.f A B f = p x + q y + r z =.f +.f x y Fgure 0. Frst order Sugeno fuzzy system th to nputs and one output. The Sugeno fuzzy system can be represented by the ANFYS equvalent system shon n Fgure. Each one of the layer (j) performs smlar operatons for each node (). The algorthm to calculate the output, S j, for each layer s as follos: Layer : Each node on ths layer s adaptve hose output s the value of the pertnence functons of all fuzzy sets. S, = µ A( x ), for =,, or S, = µ B ( x ), for = 3, 4.

8 The follong bell shape functon can be parameterzed to represent the fuzzy set, usng adaptve values a, b and c (premses): µ A ( x) = b x c + a Layer : The output of each node s the product of the nput sgnals at ths layer, resultng the eghs : S, = = µ ( x ). µ ( y), for =,. A B Layer 3: The eghts, obtaned at the prevous layer, are than normalzed: S 3, = = for =,. +, Layer 4: The nodes n ths layer are adaptve, usng the consequent parameters p, q and r. The output s calculated by the follong functon: S 4, =. f =.( p x + q y + r ), for,. = Layer 5: There s only one node n ths layer. Its output s the sum of all nput sgnals at ths layer: S,j = z =. f, for,. = Layer Layer Layer 3 Layer 4 Layer 5 x y A x A N.f Σ z y B B N.f Fgure. ANFYS equvalent archtecture. x y To solve ths problem t s necessary to determne the nonlnear premses parameters (a, b, c) and the lnear consequence parameters (p, q, r). Intally, the premse parameters are fxed, and the output z s calculated by the follong equaton:.( p x + q y + r ) +.( p x + q y r ) z = f + f =.f +.f = The hybrd learnng process conssts of a frst forard step here the premse parameters are set fxed and the netork s calculated up to the fourth layer. The consequence parameters are estmated by a mean square procedure, and the output error can be evaluated. A second step s performed backpropagatng the error, th the fxed values for the consequence parameters. The premse parameters are than updated, usng a step descent gradent technque. Ths procedure s smlar to that used to tranng a backpropagaton neural netork (Texera, 00). Ths method as appled to the forces data sets obtaned by the numercal smulatons of the fnte element model, To neural-fuzzy models ere obtaned, one for the dampng force and the other for the force appled to the valve core. In each case, the nputs are Vx and GAP and the output s the force value. Fve gaussan pertnence functons th 5 rules ere used for the nputs. Ths confguraton represents an optmzaton problem th 75 lnear and 0 nonlnear

9 parameters. Half of the data set as used n the tranng process and the nputs ere normalzed beteen and. The remanng data as used to valdate the models. The mean square error of all 500 data ponts are calculated, resultng 0.06 N for the force at the valve core, and 0.57 N for the dampng force. Fgures and 3 present the forces estmated by the neural-fuzzy models. Valve Core Force [ N ] Fgure. Neural-fuzzy model for the Force at the valve core. Fgure 3. Neural-fuzzy model for the dampng Force. The non determnstc model errors are loer than those obtaned by the analytcal model of Equaton, manly for values of Vx < 0.00 m/s. Based on ths technque, the forces drect model as detemned. In other ords, gven a GAP and Vx, t s possble to estmate both dampng and valve core forces. Hoever, nverse models are dffcult to be determned hen the tranng data has ambguty the tranng becomes mpossble. Ths s the case of GAP estmaton from the force and nput velocty values. Then, the analytcal model can be used to solve the nverse model.

10 5. Concluson The proposed desgn of ths dampng devce has promsng possbltes as a vbraton absorber. It has smple and cheap constructon and operaton compared to the conventonal devces that use hydraulc cylnders. Even for passve dampers, ths desgn can produce hgh dampng forces, thout any expend of external energy. The physcal lmts are related to the mechancal resstance of the bellos materal. The smulaton of the fnte element model ndcate that large dampng force varaton can obtaned th small dsplacements of the valve core, as requred by pezoelectrc actuators. For the analyzed vbraton relatve velocty range the flud flo could be consdered as lamnar, snce the Reynolds Number are lo at all model regons. The dampng force global behavor ndcates a lnear dependence on the nput velocty and exponental th the valve poston. Ths fact s sgnfcant to the desgn of an actve devce, snce small changes n the valve aperture provdes large varatons on the produced force. The reduced analytcal model for the dampng force and for the force at the valve core can be appled to the dynamc model of the devce coupled to a mechancal system, snce the errors on the estmated forces are sgnfcant only for very lo values of Vx. Ths approach produces a drcct and nverse model th hgh computatonal performance. The neural-fuzzy model produces loer errors on the estmates of the forces and the ncreasng of computatonal effort requred durng the operaton of a controlled system s margnal, but t can not solve the nverse problem that requres teh estmaton of the GAP from the force and nput velocty. A mxed reduced model can be defned. The forces are estmated by the neural-fuzzy model and the GAP estmaton s done by the analytcal model. Ths ay the controlled system can operate th hgh computatonal effcency th mnmum error on forces or GAP estmates. The second part of ths paper presents the smulaton of a quarter vehcle model that ncludes ths actve damper on ts suspenson. An optmum quadratc regulator assocated th a PID controller for the pezoelectrc actuator s desgned to control system vbratons. 6. References Fegel, H.J. e Romano, N., 996, Ne valve technology for actve suspenson, SAE (96077). Glomee, C.L. and Els, P.S., 998, Sem-actve hydropneumatc sprng and damper system. Journal of Terramechancs 35, pp Hagopan, J. Der, Gaudller, L. and Mallard, 999, Herarchcal control of hydraulc actve suspensons of a fast allterran mltary vehcle. Journal of Sound and Vbraton (5), Artcle No , avalable on lne at Hyun-Ung Oh and Junjro Onoda, 00, An expermental study of a semactve magneto-rheologcal flud varable damper for vbraton suppresson of truss structures. Insttute of physcs publshng: Smart Materals and Structures, pp 56 6, avalable onlne at stacks.op.org/sms//56. Jang, J.S., 993, ANFIS Adaptve-netork-based fuzzy nference system, IEEE Transacton on System Man and Cybernetcs, Vol. 3, No 3, pp Ktchng, K.J., Cole, K.J., Cebon,D., 998, Performance of a sem-actve damper for heavy vehcles, submtted to ASME Journal of Dynamc System Measuremente and Control, June. Launder, B.E. & Spaldng, D.B., 974, The Numercal Computaton of Turbulent Flos, Computer Methods n Appled Mechancs and Engneerng, vol. 3, pp Parker, G. A. e Lau, K.S., 988, A novel valve for sem-actve suspenson systems. ImechE, (C47/88): pp Texera, R.L., 00, Uma metodologa de projeto de controladores híbrdos ntelgentes com aplcações no controle atvo de vbrações mecâncas, Dssertação de Mestrado, Unversdade Federal de Uberlânda MG. 7. Copyrght Notce The author s the only responsble for the prnted materal ncluded n hs paper.