Research Article Vibration Analysis and Experimental Research of the Linear-Motor-Driven Water Piston Pump Used in the Naval Ship

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1 Research Article Vibratio Aalysis ad Experimetal Research of the Liear-Motor-Drive Water Pisto Pump Used i the Naval Ship Ye-qig Huag, Sog-li Nie, Hui Ji, ad Shuag Nie 2 Beijig Key Laboratory of Advaced Maufacturig Techology, Beijig Uiversity of Techology, Beijig 24, Chia 2 Faculty of Applied Sciece ad Egieerig, Uiversity of Toroto, Toroto, ON, Caada M5S A4 Correspodece should be addressed to Sog-li Nie; iesogli@bjut.edu.c Received 3 March 26; Accepted 22 Jue 26 Academic Editor: Carlo Raiieri Copyright 26 Ye-qig Huag et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Aimig at the existig problems of traditioal water pisto pump used i the aval ship, such as low efficiecy, high oise, large vibratio, ad oitelliget cotrol, a ew type of liear-motor-drive water pisto pump is developed ad its vibratio characteristics are aalyzed i this research. Based o the 3D model of the structure, the simulatio aalyses icludig static stress aalysis, modal aalysis, ad harmoic respose aalysis are coducted. The simulatio results reveal that the mode shape uder low frequecy stage is maily associated with the eccetricity swig of the pisto rod. The vibratio experimet results show that the resoace frequecy of liear-motor-drive water pisto pump is cocetrated upo 5 Hz ad 8 Hz i the low frequecy rage. The dampers ca chage the resoace frequecy of the system to a certai extet. The vibratio uder triagular motio curve is much better tha that of S curve, which is cosistet with the simulatio coclusio. This research provides a effective method to detect the vibratio characteristics ad a referece for desig ad optimizatio of the liear-motor-drive water pisto pump.. Itroductio High-pressure water pump is oe of importat piece of equipmet i ship egieerig [, 2]. There are several kids of high-pressure water pumps istalled i cabis, such as bilge pump, draiage pump, seawater coolig pump, highpressure water mist pump, ad desaliatio uit pump. Water axial pisto pump is characterized by higher pressure, higher volumetric efficiecy, ad lower pv values of the frictio pairs i compariso with hydraulic gear ad vae pumps [3 5]. However, the traditioal axial pisto pump drive by rotary motor has some itrisic disadvatages. For example, it would produce the alteratig axial impact forcesactigothecylideradpistowhethehigh ad low pressure are switchig ad spread out through the swash plate ad bearig. Owig to both rotary motio ad reciprocatig motio existig i covetioal axial pisto pump, they lead to a large amplitude ad low frequecy pressure pulsatio. The pulsatig force acts o the fluid ot olyithepipebutalsoithepumpbody,whichwould cause the axial rotatio ad lateral vibratio of the pisto pump. By aalyzig mechaical disturbig force, it shows that the first-order overturig momet, first-order reciprocatig iertia force, secod-order reciprocatig iertia momet, ad cetrifugal force exist i the pump set. The ubalace force produced by the mechaical motio ad the effect of fluid pulsatio would cause the fudametal frequecy oise, secod ad third times harmoic oise. Those harmoic oises have aggregated most of the pisto pump oise eergy ad determied a high total oise level. I order to overcome these shortcomigs, the liear-motor-drive water pisto pump is developed [6, 7]. Compared with the traditioal pisto pump which ca oly regulate the rotor speed of the pump, the liear-motor-drive water pisto pump ca obtai a good flow output ad vibratio performace through chagig its movemet frequecy. Previously, several researchers studied the vibratio characteristics of hydraulic pisto pump/motor. Shi [8] ivestigated the dyamic behavior of the cylider pressure cosiderig a geeral system of a pisto pump, a fluid power lie, ad ed resistace. It was show that the harmoic of maximumpulsatioamplitudewasrelatedtotherotatig

2 2 Shock ad Vibratio frequecy ad discharge pressure of the pump. There have bee efforts o uderstadig the vibratio characteristics of axial pisto pump ad motor. Bahr et al. [9] studied the vibratio characteristics of the pumpig mechaism of costat power regulated swash plate axial pisto pump by developig a mathematical model. Che et al. [ 2] studied the dyamic vibratio characteristics of the water fluid power pisto motor. They also preseted several methods for modelig ad aalyzig the vibratio sigals to diagose the faults of water fluid power pisto motor. Johasso et al. [3] studied oise characteristics which ifluece by the cross-agle i a axial pisto pump. The experimets have verified that with the optimized cross-agle, the soud level is effectively reduced. Achte [4] studied the vibratio characteristics of a variable displacemet axial pisto pump. The vibratig movemet of the swash plate is measured ad the experimets have show the effects betwee swash plate vibratio ad the displacemet of idividual pistos. Actually, the covetioal rotary-motor-drive pisto pump would produce ubalaced loadig force ad momet o the axis ad cylider body ad is proe to causig large mechaical vibratio ad oise. With the rapid developmet of liear motor techology, more ad more researchers have paid attetio to the applicatio of liear motor i the field of fluid trasmissio ad cotrol field. Early, the liear motor was used i artificial heart, sigle pisto pump, pumpig uit, air compressor, ad so o. Yamada et al. [5] developed a artificial heart drive by liear pulse motor. The thrust/iput power ratio of liear pulse motor was reached, 23.3 N/W, ad the refreshed the liear pulse motor world record at that time. Mei ad Goodall [6] preseted a subsea hydraulic pump system that the overall system cosists of two siglestage double-actio reciprocatig pumps, each of which is drive by two parallel coected double sided permaet maget liear sychroous motors. Fully digital cotrol had bee realized ad implemeted i the simulatio ad also a umber of cotrol strategies had bee tested. The results demostrate that the steady state error of the liear motor is small ad has good robustess agaist ucertaity. Zhag ad Yag [7] studied the costat flow output workig priciple of double-actio liear-motor-drive reciprocatig pump ad discuss the ifluece of motio iterval of motors ad lag agle of pump valve o the pulsatio of systemic flow rates. Hou et al. [8] ivestigated the motio characteristics of three sigle-actio ad double-actio liear motors drive reciprocatig pump i theory. A motio law of uiform-acceleratio, uiform velocity, ad uiformdeceleratio with phase differece of 6 or 2 is proposed to achieve theoretical costat flow rates. The developmet ad applicatio of smart materials also greatly promoted the developmet of the liear motor, which itegrated the techiques of liear motor itelliget cotrol ad hydraulic pump desig to achieve high efficiecy ad power desity. Xiaoetal.[9]developedagreatthrustliearmotorpumpig uit to apply i the oil field. With the oe-year ruig test, the results demostrate that the system ca accurately cotrol the output power of the liear motor through itelliget cotrol techology Cylider block : coil of liear motor 2: delivery valve 3: suctio valve 4: ilet of FDLMP 5: filtrator Cylider block 6: tak 7: pisto rod 8: active cell 9: cable : cotrol system Cotrol system : relief valve 2: outlet of FDLMP 3: flow chael 4: pisto chamber Figure : Schematic diagram of liear-motor-drive pisto pump system. Aimig at the existig problems of traditioal pisto pump used i the aval ship, such as low efficiecy, high oise, large vibratio, ad oitelliget cotrol, a ew type of liear-motor-drive water pisto pump is developed ad its vibratio characteristics are aalyzed i this research. Basedothe3Dmodelofthestructure,thesimulatio aalysis icludig static stress aalysis, modal aalysis, ad harmoic respose aalysis will be coducted, ad the the stress diagram, the atural frequecy, ad the harmoic respose spectrum of the pump system ca be obtaied. I order to optimize the liear motor motio, the motio characteristics uder differet motio curves ad modes are explored. The vibratio respose characteristics of liearmotor-drive water pisto pump will be ivestigated so as to avoid the resoace ad reduce the system vibratio. 2. Cofiguratio ad Simulatio Aalysis The developed liear-motor-drive water pisto pump is composed of four-group permaet maget liear sychroous motor, two sets of highly itegrated valve type pistopumpuits,adoesetofservocotrolsystem.the schematic diagram of liear-motor-drive water pisto pump isillustratedasifigure.theliearmotoractivecellcouples with pisto by the self-aligig mechaism ad drives the pisto rod to reciprocate cotiuously with high frequecy adhighspeed.forexample,whethepistorodisdrive by double-actio liear motor reciprocate i certai velocity plaig, oe pisto chamber (the left chamber) is i suctio process, ad the low pressure water goes through the suctio valve to the pisto chamber; meawhile the other pisto chamber (the right chamber) is i discharge process, ad the water is compressed to high pressure ad output from the delivery valve.

3 Shock ad Vibratio 3 Figure 3: Simplified 3D model of hydraulic pisto pump sectio. Figure 2: Liear-motor-drive water pisto pump system. Theoretically, the costat flow output of the liearmotor-drive water pisto pump could effectively relieve the flowpulsatioadpressureimpactadlowerthevibratioof thepump.toachievecostatflowrate,theprecisevelocity property of sigle liear motor ad the sychroizatio of multiliear motors are of sigificace. Therefore, four axes motio cotrol strategy should be specifically ivestigated basedooperatioplaadumericalsimulatiotocoordiate the operatio phases of liear motors. Furthermore,thepistopumpolyhasoefrictiopair, which ca improve the volumetric efficiecy ad is drive directly by liear motor, which ca avoid the itermediate trasmissio mechaism ad icrease the mechaical efficiecy. By meas of the servo cotrol system, the liearmotor-drive water pisto pump ca get good vibratio characteristics ad a soud flow output performace i compariso with the traditioal water pisto pump which ca oly regulate the movemet by meas of rotor speed cotrol. 2.. Static Aalysis. The liear-motor-drive water pisto pump (as show i Figure 2) icludes liear motors, water hydraulic pisto pump sectio, ad servo cotrol system. I order to reduce the mechaical vibratio of the pump system, the static simulatio aalysis is used to evaluate the stresses, strais, displacemets, ad forces i the mechaical compoets of the pump system. The water hydraulic pisto pump sectio is comprised of two large cyliders, eight distributig valves, ad eight pistos. The simplified 3D model of the water hydraulic pisto pump is built by the software SolidWorks. As show i Figure 3, without affectig the static aalysis results of mai structure, the liear motor ad the other uecessary compoets are igored i the simplified 3D model. The material of cylider block is 36L stailess steel, ad the pisto sleeve is made of alumium broze. Obviously, it is easy to obtai the physical properties of these materials such as Youg s modulus, Poisso s ratio, bulk modulus, ad shear modulus. I the case of ormal operatio coditio, the rated pressure of liear-motor-drive water pisto pump is 6 MPa ad the thrust output of each liear motor reaches 5 KN. TheresultsofstaticaalysisareillustratediFigure4.It ca be see from Figure 4 that the maximum deformatio is close to.5 mm which is far less tha the dimesios of the mechaical structure, ad it occurs at the joit betwee liear motor ad pisto. Similarly, the maximum equivalet stress is about MPa, which is also lower tha the yield stregth of the material. It meas that the mechaical structure of the pisto pump ca sustai ormal operatio loadig coditio Modal Aalysis. Modal is the atural characteristics of the mechaical structure. Each modal has its special characteristics icludig atural frequecy, mode shape, ad dampig ratio [2]. Modal aalysis is geerally employed to deal with the vibratio respose of mechaical structure, ad the modal parameters ca be obtaied by calculatio ortestaalysis.bythemodalaalysismethodtofigure out each order modal characteristic of the structure i the susceptible frequecy rage, it is possible to predict the actual vibratio respose uder actio of various exteral or iteral vibratio sources. Geerally, for a relatively simple system, the modal parameters ad system respose could be obtaied by mathematical method; for a complex system studied i this paper, excessive simplificatio will lead to the results of calculatio ot beig cosistet with actual. For the complex vibratio aalysis, may methods such as fiite elemet aalysis, operatioal modal aalysis, ad experimetal modal aalysis could be applied. While i traditioal experimetal modal aalysis, the forces excitig test sample is cotrolled, ad thetestigiscoductedithelaboratory.ioperatioal modal aalysis, the forces are just the oes which are aturally preseted durig the operatio of the structure [2]. However, either the operatioal modal aalysis method or the experimetal modal aalysis method eeds a large amout test equipmet ad experimetal data as a foudatio [22]. Therefore, the fiite elemet aalysis method is adopted i this study to obtai the system atural frequecies ad mode shapes. For a multidegree of freedom forced vibratio system, the motio equatio ca be expressed as M { χ} +C{ χ} +K{χ} = {F (t)}. ()

4 4 Shock ad Vibratio (a) Stress diagram A: static structural Equivalet stress Type: equivalet (vo-mises) stress Uit: MPa Time: max mi (b) Deformatio diagram A: static structural Total deformatio Time:.542e max.34596e.7772e.947e e e e e e 2.e Figure 4: Stress ad deformatio diagram. With the coordiate trasformatio {χ} = [u]{q}, substitute it ito (), where M{ q} + C p { q} + K{q}=u T {F (t)}, (2) where C p is the modal dampig matrix. Ithecaseofproportioaldampig,thereisC=αM+ βk,sothemodaldampigmatrixcabewritteas C p =u T [αm + βk] u = αu T Mu + βu T Ku =αm+βk. Substitutio of (3) ito (2): M r q r +(αm r +βk r ) q r +K r q r =N r (t) (r=,2,...,), where N r (t) = u T {F(t)}. Make the rth-order modal dampig C r = αm r +βk r, substitutig ito (4): q r +2ζ r ω r q r +ω 2 r q r = N r (t) M r (r=,2,...,), (5) where ζ is dampig ratio ad defied as dimesioless quatity C ζ r = r = αm r +βk r = α+βω r. (6) 2M r ω r 2M r ω r 2ω r I the case of oproportioal dampig, the key problem of (2) is the diagoalizatio of C p ; otherwise, the coordiate trasformatio equatio is still very difficult to solve. Geeral egieerig problems ofte difficultly meet the requiremets. Uder the coditio of satisfyig the egieerig precisio, it is assumed that the modal dampig matrix ca be tured ito a diagoal matrix. The exteral force actig o the system is expressed as (3) (4) {F (t)} = {F} si ωt. (7) The exteral force ca be represeted i the plural form as F (t) =Fsi ωt = Fe jωt, u T F (t) =Ne jωt. (8) Substitutio of (8) ito (2): M q +c q +c 2 q 2 + +c q +K q =N e jωt. M q +c q +c 2 q 2 + +c q +K q =N e jωt. Assumig that the atural frequecies are ot very close ad the dampig is small, the () if ay exteral force frequecy is ot close to atural frequecy, the mai role is the iertia force ad the exteral force; the dampig force ca be eglected; (2) if the frequecy of exteral force is close to the rth-order atural frequecy, the rth equatio is writte as M r q r +c r q +c r2 q 2 + +c rr q r +c r q +K r q r () =N r e jωt. Because the geeralized velocity q r at this time is much larger tha that of other geeralized velocities, the other dampig forcescabeeglected.c rr q r is the oly oe dampig term which eeds to be cosidered (c rr is a diagoal term). The viscous dampig model is a approximate treatmet method, igorig the ifluece of the odiagoal etries which are equal to icrease the respose. Thus, this kid of treatmet method should be safe. Back to (2), assumig that the dampig matrix is diagoal matrix, it could be writte as (9) M i q i +C i q i +K i q i =u T i Fejωt (i=,2,...,). () The coordiates of geeralized coordiate system q i are expressed as q i =Q i e jωt. (2) Substitutio of (2) ito (): ( ω 2 M i +jωc i +K i )Q i e jωt =u T i Fejωt. (3) Ad it could be obtaied that Q i = u T i F u T i K i ω 2 = F +jω i C i K i [( ri 2)+2jζ ir i ], (4) where r i =ω/ω i is frequecy ratio ad ζ i =C i /2 K i M i is dampig ratio.

5 Shock ad Vibratio 5 Therefore, the respose expressio of the system ca be obtaied: { χ=uq=[{u},{u 2 },...,{u} ] u T i = Fu ie jωt K i [( ri 2)+j2ζ ir i ]. i= { { q q 2. q } = } } i= q i {u} i Whe the system is oly excited at the jth coordiate, (5) u ji F j u i e jωt χ= K i [( ri 2)+j2ζ ir i ], (6) i= where u ji represets the value of jth coordiates correspodig to the ith-order mode shape. Similarly,theresposeofaycoordiatesχ k is expressed as u ji u ki F j e jωt χ k = K i [( ri 2)+j2ζ ir i ]. (7) i= Equatio (7) uses modal parameters to describe the vibratio respose, ad the real respose ca be obtaied by solvig the imagiary part. Because of that, ( r 2 i )+j2ζ ir i = where φ i =tg (2ζ i r i /( r 2 i )). Substitutio ito (7): 2 ( ri ) j2ζ ir i ( r 2 i )2 +4ζ 2 i r2 i = e jφ i, ( ri 2)2 +4ζi 2r2 i (8) u ji u ki F j e j(ωt φ i) χ k =, (9) i= K i ( ri 2)2 +4ζi 2r2 i where φ i =tg (2ζ i r i /( r 2 i )), ad the imagiary part could be writte as u ji u ki F j si (ωt φ i ) i= K i ( ri 2)2 +4ζi 2r2 i =χ k si (ωt φ i ). (2) Cosiderig that the ifluece of high-order modal is smaller tha low-order oe, the fore te-order modal is the focus of this research ad the modal aalysis will be coducted by fiite elemet calculatio based o the theory aalysis metioed above. As the water pisto pump sectio bears certai loadig uder ormal operatig coditio, the prestress modal aalysis could be more precisely compared Table : The fore te modes of vibratio. Modes Frequecy (Hz) Modal deformatio (mm) with the free modal aalysis. By modal extractio method of Block Laczos (as show i Figure 5), the first to teth atural frequecies ad mode shapes of hydraulic pisto pump sectio are gaied. Table shows the fore te-order modal frequecies ad correspodig deformatio values of each order atural frequecy. From the modal aalysis results, it is demostrated that the first-order atural frequecy is close to 66.9 Hz, which is larger tha the system motio frequecy (3 Hz), ad the modal deformatio is about 8.3 mm. Due to the large body mass ad low atural frequecy of the cylider block, the resoace of low frequecy stage easily occurs at the joit betwee the liear motor ad pisto. From Figure 5, it ca be see that the mode shape uder low frequecy stage is maily associated with the eccetricity swig of the pisto rod ad the maximum modal deformatio is withi the acceptable rage. The eccetricity swig of pisto rod would cause the ubalaced force ad vibratio as well as the wear failure of the pisto. Thus, to avoid the eccetricity swig of pisto rod, the joit desig should be optimized Harmoic Aalysis. Harmoic respose aalysis (icludig frequecy respose aalysis ad frequecy sweep aalysis) is maily used to aalyze the steady state respose of liear structure accordig to the harmoic loadig. The vibratio harmoic respose aalysis method could reveal the relatioships betwee the frequecy, displacemet, velocity, ad acceleratio uder differet frequecy [23]. I order to evade the resoace ad fatigue, avoidig the excitatio frequecy is oe of effective ways. The iput load of the harmoic respose aalysis is a siusoidal load which is chagig by the time. The primary characteristic values of the load are frequecy ad amplitude, while the load form ca be defied as force, pressure, ad the displacemet. Geerally, the simulatio results are usually the displacemet, stress, ad strai. By aalyzig the output curve, the peak respose frequecy ad amplitude ca be obtaied, which ca be used as the basis of the vibratio mechaism aalysis ad vibratio reductio desig. I a typical multiple degree of freedom system, the dyamicequatioisgiveby M { x} +C{ x} +K{x} =F(t), (2)

6 6 Shock ad Vibratio Total deformatio 2 Frequecy: Hz 4.8 max Total deformatio Frequecy: 66.9 Hz max (a) The st-order vibratio mode diagram (b) The 2d-order vibratio mode diagram Total deformatio 4 Frequecy: Hz max Total deformatio 3 Frequecy: 66.5 Hz 7.3 max (c) The 3rd-order vibratio mode diagram (d) The 4th-order vibratio mode diagram Total deformatio 5 Frequecy: Hz max (e) The 5th-order vibratio mode diagram Total deformatio 6 Frequecy: Hz 8.57 max (f) The 6th-order vibratio mode diagram Total deformatio 7 Frequecy: Hz max (g) The 7th-order vibratio mode diagram Total deformatio 8 Frequecy: Hz max (h) The 8th-order vibratio mode diagram Total deformatio Frequecy: Hz 44.8 max Total deformatio 9 Frequecy: Hz max (i) The 9th-order vibratio mode diagram (j) The th-order vibratio mode diagram Figure 5: The fore te-order mode shapes ad atural frequecies.

7 Shock ad Vibratio Deformatio (mm).2. Deformatio (mm) Frequecy (Hz) Frequecy (Hz) Axial directio (a) Deformatio of cylider wall i axial directio Radial directio (b) Deformatio of cylider wall i radial directio..8 Deformatio (mm) Frequecy (Hz) Vertical directio (c) Deformatio of cylider wall i vertical directio Figure 6: Deformatio-frequecy diagrams. where M is the mass matrix, C is the dampig matrix, K is the stiffess matrix, { x} is the acceleratio vector, { x} is the velocity vector, {x} is the displacemet vector, ad F is the load vector. For differet research purposes, it ca be used to solve the differet problem based o formula (2). For istace, whe assumig that the load vector ad dampig matrix is zero, the eigevalue of formula (2) is obtaied ad accordig to the eigevalue the vibratio mode ad atural frequecy ca be aalyzed, while the harmoic aalysis is solvig the structure respose at the assumptio that the iput load is harmoic circulatio which meas F(t) = F si(ωt). I this research, mode superpositio method is applied for harmoic aalysis of the liear-motor-drive water pisto pump based o ANSYS Workbech. The amplitude of excitig force is close to the liear motor thrust output force, amely, 5 KN, while the frequecy rage is from to 25 Hz. The cylider wall is selected as the study surface which is coveiet for the followig experimetal verificatio. The simulatio results are show i Figure 6. The three figures preset the modal deformatio of axial directio (the motio directio of liear motor), radial directio (perpedicular to the liear motor motio directio i the horizotal plae), ad vertical directio, respectively. From Figure 6, it ca be see that the maximum modal deformatio occurs i the radial directio ad catastrophic failure is ot obverse i the structure. The bottom of cylider is set as fixed support i simulatio; the vibratio amplitude is smaller i the vertical directio. I the case that the siusoidal excitatio force is 5 KN, the maximum modal deformatio is.33mm, ad the phase agle is 63. degrees. The maximum stress is about 33.3 MPa occurrig i the vertical

8 8 Shock ad Vibratio directio, ad it does ot exceed the yield stregth of the material. O the other had, the simulatio results demostrate that the resoace frequecies of the liear-motor-drive water pisto pump maily cocetrated uder 5 8 Hz. Because the motio frequecy of the pump is about 3 Hz ad the mass is large, the resoace does ot easily occur uder the low frequecy stage. Thus, whe debuggig the parameters of liear motor, the resoace frequecy should also be avoided. The harmoic aalysis provides a method to predict the dyamic characteristic of structure, which ca help to overcome the harmful effects caused by fatigue, resoace, ad other forced vibratio. Before the system vibratio testig, the PID parameter adjustmet of liear motor should be carried out. I the ext sectio, the experimet will be coducted to verify the harmoic respose aalysis. 3. Vibratio Test ad Results Aalysis 3.. Parameters Debuggig of the System. The vibratio characteristics of the liear-motor-drive water pisto pump are affected by several factors, such as mechaical structure ad servo cotrol system. Compared with the sigle cotrol mode oftraditioalpistopump,theservocotrolsystemofthe liear-motor-drive water pisto pump is able to perform a variety of servo cotrol modes. The splie (i.e., splie iterpolatio algorithm) ad PVT (i.e., locatio-time iterpolatio algorithm) motio mode are the most widely used motio modes of the liear motor. I the splie motio mode, the movemet distace is divided ito equal segmets by the time ad oly eeds to defie the positio of the coordiate poits at the time of operatio. The advatage of this method is that itcareducetheworkloadadfacilitatethecalculatig.the PVTmotiomodeeedstodefiethedistace,velocity,ad move time at the ed of each step. Therefore, it has higher request for computig performace of the servo cotrol system i compariso with the splie motio mode. Sice the pisto is directly coected with the liear motor, the istataeous flow rate of the system is proportioal to the pisto istataeous effective velocity. Thus, the property parameters of liear motor, such as the followig errors, will evetually affect the performace of liear-motordrive water pisto pump. I order to make the pump get steady flow output ad achieve the goal of low vibratio ad oise, it should set the servo cotrol characteristic parameters of liear motor appropriately. The ifluecig factors of servo cotrol characteristics iclude ot oly the motio mode but also the movemet curve. Usually the motio curves are triagular wave ad S wave curve. The velocity mathematic model of triagle wave plaig i the time domai of [, T] is described as follows: 4A T t t T 4 g (t) = { 4A T t+2a T 4 <t 3T 4 { 4A { T t 4A 3T 4 <t T, where A is the speed amplitude of liear motor. (22) Table 2: The followig errors uder differet modes ad curves. Maximum followig errors (cts) Average followig errors (cts) PVT/triagle curve PVT/S curve Splie/triagle curve Splie/S curve Note: cts =.5 μm. The velocity mathematic model of S wave plaig i the time domai of [, T/2] is give by g (t) 2 Jt2 t T V +a(t T ) T t T 2 { V 2 +a(t T 2 ) = 2 J(t T 2) 2 T 2 t T 3 V 3 2 J(t T 3) 2 T 3 t T 4 V 4 a(t T 4 ) T 4 t T 5 { V { 5 a(t T 5 )+ 2 J(t T 5) 2 T 5 t T 6, (23) where a ad J are the costats ad V, V 2, V 3, V 4,adV 5 are the speeds of each iterval, respectively. Oce the acceleratio time, uiform motio time, oe-way movemet distace, ad motio period are determied, the whole motio process ca be derived. For the workig coditio of the liear-motordrive water pisto pump, whe the four-group liear motor move at the phase differece of 9 degrees, respectively, it couldreachthecostatflowoutputtheoretically. For the sake of obtaiig the best ruig state, the dyamic respose characteristics of the liear motor uder differet motio modes ad motio curves have bee coducted i the simulatio system. The debuggig results of the liear motor i simulatio mode uder differet motio modesadcurvesareshowifigure7. I the coordiate system, the horizotal axis represets timeadtheuitisthesecod,whilethelogitudialaxes represet the velocity ad followig error, ad the measurig uitsarects/sadcts(ctsisthecoutiguitofgratig, ad it depeds o the resolutio of gratig; here oe cts is equal to.5μm), respectively. From the debuggig results (as show i Table 2), it ca be see that the simulatio results are similar ad ca meet the ormal operatio uder differet coditios. The better workig coditio is the triagle curve uder PVT motio mode. The average followig error is close to 5. cts ad the maximum followig error is 48. cts. The followig error of liear motor uder triagle curve is lesser tha that of S curve. Obviously, appropriate motio mode ad curve could effectively reduce the followig error to cotrol the actual motio track precisely ad the to lower the vibratio of the pump.

9 Shock ad Vibratio Time (s) Time (s) Mtr Act Vel (left axis) Mtr Act Pos (left axis) Mtr Fol Err (right axis) Mtr Act Vel (left axis) Mtr Act Pos (left axis) Mtr Fol Err (right axis) (a) Triagle velocity curves of liear motor at PVT mode Velocity (CTS/s) Velocity (CTS/s) 3 Followig errors (CTS) (b) S velocity curves of liear motor at PVT mode Time (s) Time (s) Mtr Act Vel (left axis) Mtr Act Pos (left axis) Mtr Fol Err (right axis) (c) Triagle velocity curves of liear motor at splie mode Followig errors (CTS) Velocity (CTS/s) Followig errors (CTS) 3 25 Velocity (CTS/s) Followig errors (CTS) 275 Mtr Act Vel (left axis) Mtr Act Pos (left axis) Mtr Fol Err (right axis) (d) S velocity curves of liear motor at splie mode Figure 7: Debuggig diagram uder differet motio modes ad curves Vibratio Experimet. The vibratio source of the liearmotor-drive water pisto pump maily comes from the motio of mechaism ad the flow pulsatio oise. The mai effect factors of the vibratio respose characteristics i this pump are the respose lag of distributig valve ad sychroous phase error of liear motor. Therefore, uder the certai system structure parameters, the cotrol strategy of liear motor has become the key factor for ifluecig the system vibratio characteristics. O the basis of the above simulatio aalysis, the vibratio experimet is coducted to obtai the vibratio ad oise iformatio. As show i Figure 8, the cylider wall is chose as the test object. The three chaels gather the vibratio sigal of x-axis (the motio directio of liear motor), y-axis (perpedicular to the liear motor motio directio i the horizotal plae), ad z-axis (vertical directio), respectively.

10 Shock ad Vibratio Table 3: Vibratio amplitude of each chael uder the triagle wave motio curve. Sigal chael Maximum amplitude Miimum amplitude Table 4: Vibratio amplitude of each chael uder the S wave motio curve. Sigal chael Maximum amplitude Miimum amplitude the liear motor movemet directio. The dampers mouted at the platform bottom ad cylider wall have a good effect to reduce the vibratio. From aother aspect, the base with a certai degree of flexibility will aggravate the vibratio of system. Therefore, the vibratio amplitude of umber 3 sigal chael is greater tha that of umber. From the previous parameters debuggig of simulatio system, it ca be kow that the followig errors of liear motor uder triagle curve are lesser tha that of S curve. The followig error was ultimately reflected i the vibratio amplitude of test results. Comparig the vibratio amplitudes (as show i Tables 3 ad 4), the vibratio of the triagular wave is sigificatly better tha that of S wave, ad this is cosistet with the simulatio results. By adoptig triagular motio curve uder PVT mode, the cylider vibratio amplitudewasreduced35%,6%,ad8%ix-, y-, adzaxis directio, respectively Power Spectrum Aalysis. Radom vibratio is the most commo type of vibratio, ad it should be studied whe carryig out a vibratio-proof desig. Power spectrum desity (PSD) is a effective method to describe radom vibratio. Geerally, the radom vibratio is characterized by the power spectral desity fuctio, for a time series χ(t) is defied as the Fourier trasform of the correlatio fuctio R χ (τ) [24]: G χy (ω) = 2π R χy (τ) e iωτ dτ. (24) Figure 8: Vibratio test poit. Though Coiv DASP software, the acceleratio, velocity, ad displacemet of the vibratio sigal are collected i real time. Accordig to previous dyamic characteristic simulatio aalysis of the liear motor, the vibratio respose test of triagle curve ad S curve uder PVT motio mode have bee coducted. The motio period is set as.6 secods, ad the stroke is 25 mm. Four groups of liear motor are movig at the phase differece of 9 degrees. After the calculus processig of the collected vibratio data, the amplitude distributio i the time domai of the x-, y-, adz-axis is show i Figure 9. It ca be see from Figure 9 that the periodic vibratio exists i the system ad the vibratio impact cycle is roughly idetical to the system movemet cycle. It demostrates that the mai vibratio source of the liear-motor-drive water pisto pump is cotiuous high-speed reciprocatig motio of the liear motor uit. The vibratio impact is derived from the ubalaced torque caused by the thrust of the liear motor. The vibratio amplitudes of each chael uder the S ad triagle curve are show i Tables 3 ad 4, respectively. Compared to the results of three sigal chaels, the vibratio peak appears i umber 2 sigal chael adthemaximumamplitudeoftriagleadscurveare μm ad 676.8μm, respectively. It meas that severe vibratio occurs i the directio which is perpedicular to The correlatio fuctio is used to describe the relatioship betwee the two values of the same sigal i the iterval τ ad ca be writte as T R (τ) = lim T T x (t) x (t+τ) dt, (25) where x(t) is the vibratio respose of ay trasiet t. The PSD is widely used because the modes could be idicated clearly by spectral peaks. Power spectrum is the cocept of statistical average radom process that is expressed by the sigal power with the chage of frequecy. The horizotal axis uit of power spectrum desity plot is Hz, ad the vertical axis is g 2 /Hz, where g 2 represets the mea square value of the acceleratio. With the DASP software to aalyze the collected data, the power spectrum of the liear-motordrive water pisto pump uder differet motio coditio isshowifigureadthepowerspectrumsofeachsigal chael are displayed i differet graph. Compared to the graphs of sigal chael i the differet motio mode, it ca be observed that the tred of power spectrum is basically idetical. Like the previous aalysis, the power spectrum oly reserves the amplitude iformatio adlostthephaseiformatio.so,thediffereceliesithe amplitude at the resoace frequecy. The amplitudes of each chael uder triagular curve are.25,.82, ad.78 g 2, while the amplitude of each chael uder S curve are.6,.3, ad.36 g 2.Obviously,thevibratioamplitudeoftheS wave is 4 6 times that of triagular wave. The experimetal data show that the vibratio uder triagular motio curve

11 Shock ad Vibratio (D) (2D) (s) 2 2 (s) (3D) (s) (a) Triagle curve, PVT motio mode (D) (s) (2D) (s) (3D) (s) (b) S curve, PVT motio mode Figure 9: Vibratio amplitude distributio i the time domai of the x-, y-,ad z-axis..5 ().5 ().4.4 (g 2 ).3.2 (g 2 ) (g 2 ) (g 2 ). 5 k 5 2k 25 (Hz) (2) k 5 2k 25 (Hz) (3) k 5 2k 25 (Hz) (a) Triagle curve, PVT motio mode (g 2 ) (g 2 ). 5 k 5 2k 25 (Hz).5 (2) k 5 2k 25 (Hz).5 (3) k 5 2k 25 (Hz) (b) S curve, PVT motio mode Figure : Power spectrum of liear-motor-drive water pisto pump system uder triagle ad S motio curve.

12 2 Shock ad Vibratio is much better tha that of S curve, which is cosistet with the previous simulatio coclusio. Differet motio curves will affect the followig errors of the liear motor, which is reflected i the vibratio of the system fially. It ca be see from Figure (a) that, uder the triagular wave motio curve, the peak of umber sigal chael is olyoe-thirdthepeaksofumber2adumber3.thepeak frequecy of umber sigal chael (the motio directio of liear motor) oly appears ear 5 Hz ad has a little differecewiththevalueprovidedbythefemaalysis.withthe dampers equipped o the cylider body i the motio directio of liear motor, it will ievitably lead to the differece betwee the simulatio results ad the experimetal data. Accordig to the data collected by umber 2 sigal chael, the resoace frequecy is maily cocetrated i the viciity of 5 Hz ad 8 Hz. Similarly, the aalysis result of umber 3 sigal chael is essetially cosistet with the harmoic aalysis result. Therefore, the resoace frequecy of liear-motor-drive water pisto pump is cocetrated upo 5Hz ad 8Hz i the low frequecy rage. The dampers ca chage the resoace frequecy of the system to a certai extet. Durig the operatio of the liear-motordrive water pisto pump, the resoace frequecy rage detected by simulatio aalysis ad experimet test should beavoidedtoeffectivelyreducethevibratio. 4. Coclusio Based o the prototype ad three-dimesioal model of the liear-motor-drive water pisto pump, this paper has carried out the vibratio characteristic aalysis. The fiite elemet aalysis software has bee employed to perform a static stress, modal, ad harmoic aalysis. The maximum deformatio ad equivalet stress of the mechaical structure were measured at the rated workig coditio. From the results preseted i this paper, the importat coclusios obtaiedcabedrawasfollows: () The simulatio results reveal that the maximum deformatio ad equivalet stress of the mechaical structure i this pump are.5 mm ad MPa, respectively, which is lower tha the yield stregth of the material ad the allowed deformatio. The first-order atural frequecy is close to 66.9 Hz, which is much larger tha the system motio frequecy (3 Hz), ad the modal deformatio is about 8.3 mm. Ad the mode shape uder low frequecy stage is associated with the eccetricity swig of the pisto rod ad the maximum deformatio is withi the acceptable rage. (2) The harmoic aalysis ad vibratio experimet results show that the resoace frequecy of liear-motordrive water pisto pump is cocetrated upo 5 Hz ad 8 Hz i the low frequecy rage. The dampers ca chage the resoace frequecy of the system to a certai extet. (3) The vibratio uder triagular motio curve is much better tha that of S curve, which is cosistet with the previous simulatio coclusio. The research provides a effective method to detect the vibratio characteristics of the liear-motor-drive water pisto pump ad also a referece for desig ad optimizatio ofthepistopump. Competig Iterests The authors declare that they have o competig iterests. Ackowledgmets The authors would like to thak the Natioal Natural Sciece Foudatios of Chia (os ad 5722), Natioal High-Tech R&D (863) Program (o. 22AA93), Beijig Natural Sciece Foudatio (o ), ad Chia Postdoctoral Sciece Foudatio (o. 25M58946) for their fudig for this research. Refereces []C.Louis,F.Michael,adG.K.Thomas, Waterhydraulic eergy savigs vehicle, i Proceedigs of the 8th Scadiavia Iteratioal Coferece o Fluid Power (SICFP 3), pp.7 28, Tampere, Filad, May 23. 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