research article vibration analysis and experimental research of … · 2019. 7. 30. · research...

Research ArticleVibration Analysis and Experimental Research of theLinear-Motor-Driven Water Piston Pump Used in the Naval Ship

Ye-qing Huang,1 Song-lin Nie,1 Hui Ji,1 and Shuang Nie2

1Beijing Key Laboratory of Advanced Manufacturing Technology, Beijing University of Technology, Beijing 100124, China2Faculty of Applied Science and Engineering, University of Toronto, Toronto, ON, Canada M5S 1A4

Correspondence should be addressed to Song-lin Nie; [email protected]

Received 3 March 2016; Accepted 22 June 2016

Academic Editor: Carlo Rainieri

Copyright © 2016 Ye-qing Huang et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Aiming at the existing problems of traditional water piston pump used in the naval ship, such as low efficiency, high noise,large vibration, and nonintelligent control, a new type of linear-motor-driven water piston pump is developed and its vibrationcharacteristics are analyzed in this research. Based on the 3D model of the structure, the simulation analyses including static stressanalysis, modal analysis, and harmonic response analysis are conducted. The simulation results reveal that the mode shape underlow frequency stage is mainly associated with the eccentricity swing of the piston rod. The vibration experiment results show thatthe resonance frequency of linear-motor-driven water piston pump is concentrated upon 500Hz and 800Hz in the low frequencyrange. The dampers can change the resonance frequency of the system to a certain extent. The vibration under triangular motioncurve is much better than that of S curve, which is consistent with the simulation conclusion. This research provides an effectivemethod to detect the vibration characteristics and a reference for design and optimization of the linear-motor-driven water pistonpump.

1. Introduction

High-pressure water pump is one of important piece ofequipment in ship engineering [1, 2]. There are several kindsof high-pressure water pumps installed in cabins, such asbilge pump, drainage pump, seawater cooling pump, high-pressure water mist pump, and desalination unit pump.Water axial piston pump is characterized by higher pressure,higher volumetric efficiency, and lower 𝑝V values of thefriction pairs in comparison with hydraulic gear and vanepumps [3–5]. However, the traditional axial piston pumpdriven by rotary motor has some intrinsic disadvantages.For example, it would produce the alternating axial impactforces acting on the cylinder and piston when the highand low pressure are switching and spread out through theswash plate and bearing. Owing to both rotary motion andreciprocating motion existing in conventional axial pistonpump, they lead to a large amplitude and low frequencypressure pulsation. The pulsating force acts on the fluid notonly in the pipe but also in the pump body, which wouldcause the axial rotation and lateral vibration of the piston

pump. By analyzingmechanical disturbing force, it shows thatthe first-order overturningmoment, first-order reciprocatinginertia force, second-order reciprocating inertia moment,and centrifugal force exist in the pump set. The unbalanceforce produced by the mechanical motion and the effect offluid pulsationwould cause the fundamental frequency noise,second and third times harmonic noise. Those harmonicnoises have aggregated most of the piston pump noise energyand determined a high total noise level. In order to overcomethese shortcomings, the linear-motor-driven water pistonpump is developed [6, 7]. Compared with the traditionalpiston pump which can only regulate the rotor speed of thepump, the linear-motor-drivenwater piston pump can obtaina good flow output and vibration performance throughchanging its movement frequency.

Previously, several researchers studied the vibrationcharacteristics of hydraulic piston pump/motor. Shin [8]investigated the dynamic behavior of the cylinder pressureconsidering a general system of a piston pump, a fluid powerline, and end resistance. It was shown that the harmonic ofmaximum pulsation amplitude was related to the rotating

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 4608328, 13 pageshttp://dx.doi.org/10.1155/2016/4608328

2 Shock and Vibration

frequency and discharge pressure of the pump. There havebeen efforts on understanding the vibration characteristicsof axial piston pump and motor. Bahr et al. [9] studiedthe vibration characteristics of the pumping mechanism ofconstant power regulated swash plate axial piston pump bydeveloping amathematical model. Chen et al. [10–12] studiedthe dynamic vibration characteristics of the water fluidpower piston motor. They also presented several methodsfor modeling and analyzing the vibration signals to diagnosethe faults of water fluid power piston motor. Johanssonet al. [13] studied noise characteristics which influence bythe cross-angle in an axial piston pump. The experimentshave verified that with the optimized cross-angle, the soundlevel is effectively reduced. Achten [14] studied the vibrationcharacteristics of a variable displacement axial piston pump.The vibrating movement of the swash plate is measured andthe experiments have shown the effects between swash platevibration and the displacement of individual pistons.

Actually, the conventional rotary-motor-driven pistonpump would produce unbalanced loading force and momenton the axis and cylinder body and is prone to causing largemechanical vibration and noise. With the rapid developmentof linear motor technology, more and more researchers havepaid attention to the application of linear motor in the fieldof fluid transmission and control field. Early, the linear motorwas used in artificial heart, single piston pump, pumping unit,air compressor, and so on. Yamada et al. [15] developed anartificial heart driven by linear pulse motor. The thrust/inputpower ratio of linear pulse motor was reached, 23.3N/W,and then refreshed the linear pulse motor world record atthat time. Mei and Goodall [16] presented a subsea hydraulicpump system that the overall system consists of two single-stage double-action reciprocating pumps, each of which isdriven by two parallel connected double sided permanentmagnet linear synchronous motors. Fully digital control hadbeen realized and implemented in the simulation and alsoa number of control strategies had been tested. The resultsdemonstrate that the steady state error of the linear motoris small and has good robustness against uncertainty. Zhangand Yang [17] studied the constant flow output workingprinciple of double-action linear-motor-driven reciprocat-ing pump and discuss the influence of motion interval ofmotors and lag angle of pump valve on the pulsation ofsystemic flow rates. Hou et al. [18] investigated the motioncharacteristics of three single-action and double-action linearmotors driven reciprocating pump in theory. A motionlaw of uniform-acceleration, uniform velocity, and uniform-deceleration with phase difference of 60∘ or 120∘ is proposedto achieve theoretical constant flow rates. The developmentand application of smart materials also greatly promotedthe development of the linear motor, which integrated thetechniques of linear motor intelligent control and hydraulicpump design to achieve high efficiency and power density.Xiao et al. [19] developed a great thrust linearmotor pumpingunit to apply in the oil field. With the one-year running test,the results demonstrate that the system can accurately controlthe output power of the linear motor through intelligentcontrol technology.

1

2

3

4

5

6

7

89

10

11

12

13

14

1: coil of linear motor2: delivery valve 3: suction valve4: inlet of FDLMP5: filtrator

6: tank7: piston rod8: active cell9: cable10: control system

11: relief valve12: outlet of FDLMP13: flow channel14: piston chamber

Cylinder blockCylinder block

Controlsystem

Figure 1: Schematic diagram of linear-motor-driven piston pumpsystem.

Aiming at the existing problems of traditional pistonpump used in the naval ship, such as low efficiency, highnoise, large vibration, and nonintelligent control, a new typeof linear-motor-driven water piston pump is developed andits vibration characteristics are analyzed in this research.Based on the 3D model of the structure, the simulationanalysis including static stress analysis, modal analysis, andharmonic response analysis will be conducted, and thenthe stress diagram, the natural frequency, and the harmonicresponse spectrum of the pump system can be obtained.In order to optimize the linear motor motion, the motioncharacteristics under different motion curves and modes areexplored. The vibration response characteristics of linear-motor-driven water piston pump will be investigated so as toavoid the resonance and reduce the system vibration.

2. Configuration and Simulation Analysis

The developed linear-motor-driven water piston pump iscomposed of four-group permanent magnet linear syn-chronous motor, two sets of highly integrated valve typepiston pump units, and one set of servo control system. Theschematic diagramof linear-motor-drivenwater piston pumpis illustrated as in Figure 1.The linearmotor active cell coupleswith piston by the self-aligning mechanism and drives thepiston rod to reciprocate continuously with high frequencyand high speed. For example, when the piston rod is drivenby double-action linear motor reciprocate in certain velocityplanning, one piston chamber (the left chamber) is in suctionprocess, and the low pressure water goes through the suctionvalve to the piston chamber; meanwhile the other pistonchamber (the right chamber) is in discharge process, and thewater is compressed to high pressure and output from thedelivery valve.

Shock and Vibration 3

Figure 2: Linear-motor-driven water piston pump system.

Theoretically, the constant flow output of the linear-motor-driven water piston pump could effectively relieve theflow pulsation and pressure impact and lower the vibration ofthe pump. To achieve constant flow rate, the precise velocityproperty of single linear motor and the synchronization ofmultilinear motors are of significance. Therefore, four axesmotion control strategy should be specifically investigatedbased on operation plan and numerical simulation to coor-dinate the operation phases of linear motors.

Furthermore, the piston pump only has one friction pair,which can improve the volumetric efficiency and is drivendirectly by linear motor, which can avoid the intermediatetransmission mechanism and increase the mechanical effi-ciency. By means of the servo control system, the linear-motor-driven water piston pump can get good vibrationcharacteristics and a sound flow output performance in com-parison with the traditional water piston pump which canonly regulate the movement by means of rotor speed control.

2.1. Static Analysis. The linear-motor-driven water pistonpump (as shown in Figure 2) includes linear motors, waterhydraulic piston pump section, and servo control system.In order to reduce the mechanical vibration of the pumpsystem, the static simulation analysis is used to evaluate thestresses, strains, displacements, and forces in the mechanicalcomponents of the pump system.

The water hydraulic piston pump section is comprisedof two large cylinders, eight distributing valves, and eightpistons. The simplified 3D model of the water hydraulicpiston pump is built by the software SolidWorks. As shownin Figure 3, without affecting the static analysis results ofmain structure, the linear motor and the other unnecessarycomponents are ignored in the simplified 3D model. Thematerial of cylinder block is 316L stainless steel, and the pistonsleeve is made of aluminum bronze. Obviously, it is easyto obtain the physical properties of these materials such asYoung’s modulus, Poisson’s ratio, bulk modulus, and shearmodulus.

In the case of normal operation condition, the ratedpressure of linear-motor-driven water piston pump is 6MPa

Figure 3: Simplified 3D model of hydraulic piston pump section.

and the thrust output of each linear motor reaches 5 KN.The results of static analysis are illustrated in Figure 4. Itcan be seen from Figure 4 that the maximum deformation isclose to 0.15mm which is far less than the dimensions of themechanical structure, and it occurs at the joint between linearmotor and piston. Similarly, themaximumequivalent stress isabout 95.398MPa, which is also lower than the yield strengthof the material. It means that the mechanical structure of thepiston pump can sustain normal operation loading condition.

2.2. Modal Analysis. Modal is the natural characteristicsof the mechanical structure. Each modal has its specialcharacteristics including natural frequency, mode shape, anddamping ratio [20]. Modal analysis is generally employedto deal with the vibration response of mechanical structure,and the modal parameters can be obtained by calculationor test analysis. By the modal analysis method to figureout each order modal characteristic of the structure in thesusceptible frequency range, it is possible to predict theactual vibration response under action of various external orinternal vibration sources.

Generally, for a relatively simple system, the modalparameters and system response could be obtained by math-ematical method; for a complex system studied in this paper,excessive simplification will lead to the results of calculationnot being consistent with actual. For the complex vibrationanalysis, many methods such as finite element analysis,operational modal analysis, and experimental modal analysiscould be applied. While in traditional experimental modalanalysis, the forces exciting test sample is controlled, andthe testing is conducted in the laboratory. In operationalmodal analysis, the forces are just the ones which arenaturally presented during the operation of the structure [21].However, either the operationalmodal analysismethod or theexperimental modal analysis method needs a large amounttest equipment and experimental data as a foundation [22].Therefore, the finite element analysis method is adopted inthis study to obtain the system natural frequencies and modeshapes.

For amultidegree of freedom forced vibration system, themotion equation can be expressed as

𝑀{��} + 𝐶 {��} + 𝐾 {𝜒} = {𝐹 (𝑡)} . (1)


A: static structuralEquivalent stressType: equivalent

Unit: MPaTime: 1

95.398 max

0.0044643 min

84.79874.19963.653.00142.40131.80221.20310.604

(von-Mises) stress

(a) Stress diagram

Unit: mmTime: 1

Total deformationType: total deformation

1.51421e − 1 max1.34596e − 11.17772e − 11.00947e − 18.41225e − 26.72980e − 25.04735e − 23.36490e − 21.68245e − 20.00000e0 min

A: static structural

(b) Deformation diagram

Figure 4: Stress and deformation diagram.

With the coordinate transformation {𝜒} = [𝑢]{𝑞}, substituteit into (1), where

𝑀{��} + 𝐶𝑝{��} + 𝐾 {𝑞} = 𝑢

𝑇

{𝐹 (𝑡)} , (2)

where 𝐶𝑝is the modal damping matrix.

In the case of proportional damping, there is 𝐶 = 𝛼𝑀 +

𝛽𝐾, so the modal damping matrix can be written as

𝐶𝑝= 𝑢𝑇

[𝛼𝑀 + 𝛽𝐾] 𝑢 = 𝛼𝑢𝑇

𝑀𝑢 + 𝛽𝑢𝑇

𝐾𝑢

= 𝛼𝑀 + 𝛽𝐾.

(3)

Substitution of (3) into (2):

𝑀𝑟��𝑟

+ (𝛼𝑀𝑟+ 𝛽𝐾𝑟) ��𝑟

+ 𝐾𝑟𝑞𝑟= 𝑁𝑟(𝑡)

(𝑟 = 1, 2, . . . , 𝑛) ,(4)

where𝑁𝑟(𝑡) = 𝑢

𝑇

{𝐹(𝑡)}.Make the rth-order modal damping 𝐶

𝑟= 𝛼𝑀

𝑟+ 𝛽𝐾𝑟,

substituting into (4):

��𝑟

+ 2𝜁𝑟𝜔𝑛𝑟��𝑟

+ 𝜔2

𝑛𝑟

𝑞𝑟=

𝑁𝑟(𝑡)

𝑀𝑟

(𝑟 = 1, 2, . . . , 𝑛) , (5)

where 𝜁 is damping ratio and defined as dimensionlessquantity

𝜁𝑟=

𝐶𝑟

2𝑀𝑟𝜔𝑛𝑟

=𝛼𝑀𝑟+ 𝛽𝐾𝑟

2𝑀𝑟𝜔𝑛𝑟

=𝛼 + 𝛽𝜔

𝑛𝑟

2𝜔𝑛𝑟

. (6)

In the case of nonproportional damping, the key problemof (2) is the diagonalization of 𝐶

𝑝; otherwise, the coordinate

transformation equation is still very difficult to solve. Generalengineering problems often difficultlymeet the requirements.Under the condition of satisfying the engineering precision, itis assumed that themodal dampingmatrix can be turned intoa diagonal matrix. The external force acting on the system isexpressed as

{𝐹 (𝑡)} = {𝐹} sin𝜔𝑡. (7)

The external force can be represented in the plural form as

𝐹 (𝑡) = 𝐹 sin𝜔𝑡 = 𝐹𝑒𝑗𝜔𝑡

,

𝑢𝑇

𝐹 (𝑡) = 𝑁𝑒𝑗𝜔𝑡

.

(8)

Substitution of (8) into (2):

𝑀1��1

+ 𝑐11��1

+ 𝑐12��2

+ ⋅ ⋅ ⋅ + 𝑐1𝑛��𝑛

+ 𝐾1𝑞1= 𝑁1𝑒𝑗𝜔𝑡

.

.

.

𝑀𝑛��𝑛

+ 𝑐𝑛1��1

+ 𝑐𝑛2��2

+ ⋅ ⋅ ⋅ + 𝑐𝑛𝑛��𝑛

+ 𝐾𝑛𝑞𝑛= 𝑁𝑛𝑒𝑗𝜔𝑡

.

(9)

Assuming that the natural frequencies are not very closeand the damping is small, then (1) if any external forcefrequency is not close to natural frequency, the main role isthe inertia force and the external force; the damping force canbe neglected; (2) if the frequency of external force is close tothe rth-order natural frequency, the rth equation is written as

𝑀𝑟��𝑟

+ 𝑐𝑟1��1

+ 𝑐𝑟2��2

+ ⋅ ⋅ ⋅ + 𝑐𝑟𝑟��𝑟

+ 𝑐𝑟𝑛��𝑛

+ 𝐾𝑟𝑞𝑟

= 𝑁𝑟𝑒𝑗𝜔𝑡

.(10)

Because the generalized velocity ��𝑟

at this time is much largerthan that of other generalized velocities, the other dampingforces can be neglected. 𝑐

𝑟𝑟��𝑟

is the only one damping termwhich needs to be considered (𝑐

𝑟𝑟is a diagonal term). The

viscous dampingmodel is an approximate treatmentmethod,ignoring the influence of the nondiagonal entries which areequal to increase the response. Thus, this kind of treatmentmethod should be safe.

Back to (2), assuming that the dampingmatrix is diagonalmatrix, it could be written as

𝑀𝑖��𝑖

+ 𝐶𝑖��𝑖

+ 𝐾𝑖𝑞𝑖= 𝑢𝑇

𝑖

𝐹𝑒𝑗𝜔𝑡

(𝑖 = 1, 2, . . . , 𝑛) . (11)

The coordinates of generalized coordinate system 𝑞𝑖are

expressed as

𝑞𝑖= 𝑄𝑖𝑒𝑗𝜔𝑡

. (12)

Substitution of (12) into (11):(−𝜔2

𝑀𝑖+ 𝑗𝜔𝐶

𝑖+ 𝐾𝑖)𝑄𝑖𝑒𝑗𝜔𝑡

= 𝑢𝑇

𝑖

𝐹𝑒𝑗𝜔𝑡

. (13)

And it could be obtained that

𝑄𝑖=

𝑢𝑇

𝑖

𝐹

𝐾𝑖− 𝜔2 + 𝑗𝜔

𝑖𝐶𝑖

=𝑢𝑇

𝑖

𝐹

𝐾𝑖[(1 − 𝑟2

𝑖

) + 2𝑗𝜁𝑖𝑟𝑖], (14)

where 𝑟𝑖= 𝜔/𝜔

𝑛𝑖is frequency ratio and 𝜁

𝑖= 𝐶𝑖/2√𝐾

𝑖𝑀𝑖is

damping ratio.


Therefore, the response expression of the system can beobtained:

𝜒 = 𝑢𝑞 = [{𝑢}1, {𝑢2} , . . . , {𝑢}

𝑛]

{{{{{{

{{{{{{

{

𝑞1

𝑞2

.

.

.

𝑞𝑛

}}}}}}

}}}}}}

}

=

𝑛

∑

𝑖=1

𝑞𝑖{𝑢}𝑖

=

𝑛

∑

𝑖=1

𝑢𝑇

𝑖

𝐹𝑢𝑖𝑒𝑗𝜔𝑡

𝐾𝑖[(1 − 𝑟2

𝑖

) + 𝑗2𝜁𝑖𝑟𝑖].

(15)

When the system is only excited at the jth coordinate,

𝜒 =

𝑛

∑

𝑖=1

𝑢𝑗𝑖𝐹𝑗𝑢𝑖𝑒𝑗𝜔𝑡

𝐾𝑖[(1 − 𝑟2

𝑖

) + 𝑗2𝜁𝑖𝑟𝑖], (16)

where 𝑢𝑗𝑖represents the value of 𝑗th coordinates correspond-

ing to the 𝑖th-order mode shape.Similarly, the response of any coordinates 𝜒

𝑘is expressed

as

𝜒𝑘=

𝑛

∑

𝑖=1

𝑢𝑗𝑖𝑢𝑘𝑖𝐹𝑗𝑒𝑗𝜔𝑡

𝐾𝑖[(1 − 𝑟2

𝑖

) + 𝑗2𝜁𝑖𝑟𝑖]. (17)

Equation (17) uses modal parameters to describe the vibra-tion response, and the real response can be obtained bysolving the imaginary part. Because of that,

1

(1 − 𝑟2𝑖

) + 𝑗2𝜁𝑖𝑟𝑖

=(1 − 𝑟

2

𝑖

) − 𝑗2𝜁𝑖𝑟𝑖

(1 − 𝑟2𝑖

)2

+ 4𝜁2𝑖

𝑟2𝑖

=1

√(1 − 𝑟2𝑖

)2

+ 4𝜁2𝑖

𝑟2𝑖

𝑒−𝑗𝜑𝑖 ,

(18)

where 𝜑𝑖= 𝑡𝑔−1

(2𝜁𝑖𝑟𝑖/(1 − 𝑟

2

𝑖

)).Substitution into (17):

𝜒𝑘=

𝑛

∑

𝑖=1

𝑢𝑗𝑖𝑢𝑘𝑖𝐹𝑗𝑒𝑗(𝜔𝑡−𝜑𝑖)

𝐾𝑖

√(1 − 𝑟2𝑖

)2

+ 4𝜁2𝑖

𝑟2𝑖

, (19)

where 𝜑𝑖= 𝑡𝑔−1

(2𝜁𝑖𝑟𝑖/(1 − 𝑟

2

𝑖

)), and the imaginary part couldbe written as

𝑛

∑

𝑖=1

𝑢𝑗𝑖𝑢𝑘𝑖𝐹𝑗

𝐾𝑖

√(1 − 𝑟2𝑖

)2

+ 4𝜁2𝑖

𝑟2𝑖

sin (𝜔𝑡 − 𝜑𝑖)

= 𝜒𝑘sin (𝜔𝑡 − 𝜑

𝑖) .

(20)

Considering that the influence of high-ordermodal is smallerthan low-order one, the fore ten-order modal is the focusof this research and the modal analysis will be conductedby finite element calculation based on the theory analysismentioned above. As the water piston pump section bearscertain loading under normal operating condition, the pre-stress modal analysis could be more precisely compared

Table 1: The fore ten modes of vibration.

Modes Frequency (Hz) Modal deformation (mm)1 166.19 18.3392 166.38 14.0803 166.50 17.3104 166.59 18.4485 166.87 17.2636 167.27 18.5707 167.46 19.2878 167.84 19.2539 380.25 43.91210 382.62 44.080

with the free modal analysis. By modal extraction methodof Block Lanczos (as shown in Figure 5), the first to tenthnatural frequencies and mode shapes of hydraulic pistonpump section are gained.

Table 1 shows the fore ten-order modal frequencies andcorresponding deformation values of each order naturalfrequency. From themodal analysis results, it is demonstratedthat the first-order natural frequency is close to 166.19Hz,which is larger than the systemmotion frequency (3Hz), andthe modal deformation is about 18.3mm. Due to the largebody mass and low natural frequency of the cylinder block,the resonance of low frequency stage easily occurs at the jointbetween the linear motor and piston. From Figure 5, it can beseen that themode shape under low frequency stage ismainlyassociated with the eccentricity swing of the piston rod andthe maximum modal deformation is within the acceptablerange. The eccentricity swing of piston rod would cause theunbalanced force and vibration as well as the wear failure ofthe piston.Thus, to avoid the eccentricity swing of piston rod,the joint design should be optimized.

2.3.HarmonicAnalysis. Harmonic response analysis (includ-ing frequency response analysis and frequency sweep anal-ysis) is mainly used to analyze the steady state responseof linear structure according to the harmonic loading. Thevibration harmonic response analysis method could revealthe relationships between the frequency, displacement, veloc-ity, and acceleration under different frequency [23].

In order to evade the resonance and fatigue, avoiding theexcitation frequency is one of effective ways.The input load ofthe harmonic response analysis is a sinusoidal load which ischanging by the time.The primary characteristic values of theload are frequency and amplitude, while the load form can bedefined as force, pressure, and the displacement. Generally,the simulation results are usually the displacement, stress,and strain. By analyzing the output curve, the peak responsefrequency and amplitude can be obtained, which can be usedas the basis of the vibrationmechanism analysis and vibrationreduction design.

In a typical multiple degree of freedom system, thedynamic equation is given by

𝑀{��} + 𝐶 {��} + 𝐾 {𝑥} = 𝐹 (𝑡) , (21)


18.339 max16.30114.26412.22610.1888.15076.1134.07532.03770 min

B: modalTotal deformationType: total deformationFrequency: 166.19 HzUnit: mm

(a) The 1st-order vibration mode diagram

B: modalTotal deformation 2Type: total deformationFrequency: 166.38 HzUnit: mm

14.08 max12.51610.9519.38687.82236.25784.69343.12891.56450 min

(b) The 2nd-order vibration mode diagram


17.31 max15.38713.46311.549.61677.69335.773.84671.92330 min

(c) The 3rd-order vibration mode diagram


18.448 max16.39814.34812.29810.2498.19896.14924.09952.04970 min

(d) The 4th-order vibration mode diagram

B: modalTotal deformation 5Type: total deformationFrequency: 166.87Unit: mm

17.263 max15.34513.42711.5099.59057.67245.75433.83621.91810 min

Hz

(e) The 5th-order vibration mode diagram


18.57 max16.50714.44412.3810.3178.25356.19014.12672.06340 min

(f) The 6th-order vibration mode diagram

B: modalTotal deformation 7Type: total deformation

Unit: mm19.287 max17.14415.00112.85810.7158.57196.42894.2862.1430 min

Frequency: 167.46 Hz

(g) The 7th-order vibration mode diagram


19.253 max17.11414.97512.83610.6968.5576.41784.27852.13930 min

(h) The 8th-order vibration mode diagram

B: modalTotal deformation 9Type: total deformation

Unit: mm43.912 max39.03334.15429.27524.39519.51614.6379.75824.87910 min

Frequency: 380.25 Hz

(i) The 9th-order vibration mode diagram


44.08 max39.18234.28429.38724.48919.59114.6939.79564.89780 min

(j) The 10th-order vibration mode diagram

Figure 5: The fore ten-order mode shapes and natural frequencies.


500 1000 1500 2000 2500

Frequency (Hz)

Axial direction

0

0.00

0.01

0.02

0.03

Def

orm

atio

n (m

m)

(a) Deformation of cylinder wall in axial direction

500 1000 1500 2000 2500

Frequency (Hz)0

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Def

orm

atio

n (m

m)

Radial direction(b) Deformation of cylinder wall in radial direction

Frequency (Hz)0 500 1000 1500 2000 2500

0.000

0.002

0.004

0.006

0.008

0.010

Def

orm

atio

n (m

m)

Vertical direction(c) Deformation of cylinder wall in vertical direction

Figure 6: Deformation-frequency diagrams.

where 𝑀 is the mass matrix, 𝐶 is the damping matrix, 𝐾 isthe stiffness matrix, {��} is the acceleration vector, {��} is thevelocity vector, {𝑥} is the displacement vector, and 𝐹 is theload vector.

For different research purposes, it can be used to solve thedifferent problem based on formula (21). For instance, whenassuming that the load vector and dampingmatrix is zero, theeigenvalue of formula (21) is obtained and according to theeigenvalue the vibration mode and natural frequency can beanalyzed, while the harmonic analysis is solving the structureresponse at the assumption that the input load is harmoniccirculation which means 𝐹(𝑡) = 𝐹

0sin(𝜔𝑡). In this research,

mode superposition method is applied for harmonic analysisof the linear-motor-driven water piston pump based onANSYS Workbench. The amplitude of exciting force is closeto the linear motor thrust output force, namely, 5 KN, while

the frequency range is from 10 to 2500Hz. The cylinder wallis selected as the study surface which is convenient for thefollowing experimental verification. The simulation resultsare shown in Figure 6.

The three figures present the modal deformation of axialdirection (themotion direction of linearmotor), radial direc-tion (perpendicular to the linear motor motion directionin the horizontal plane), and vertical direction, respectively.From Figure 6, it can be seen that the maximum modaldeformation occurs in the radial direction and catastrophicfailure is not obverse in the structure.The bottom of cylinderis set as fixed support in simulation; the vibration amplitude issmaller in the vertical direction. In the case that the sinusoidalexcitation force is 5 KN, the maximum modal deformationis 0.33mm, and the phase angle is −163.11 degrees. Themaximum stress is about 33.3MPa occurring in the vertical


direction, and it does not exceed the yield strength of thematerial.

On the other hand, the simulation results demonstratethat the resonance frequencies of the linear-motor-drivenwater piston pump mainly concentrated under 500–800Hz.Because the motion frequency of the pump is about 3Hzand the mass is large, the resonance does not easily occurunder the low frequency stage. Thus, when debugging theparameters of linear motor, the resonance frequency shouldalso be avoided. The harmonic analysis provides a methodto predict the dynamic characteristic of structure, which canhelp to overcome the harmful effects caused by fatigue, reso-nance, and other forced vibration. Before the systemvibrationtesting, the PID parameter adjustment of linearmotor shouldbe carried out. In the next section, the experiment will beconducted to verify the harmonic response analysis.

3. Vibration Test and Results Analysis

3.1. Parameters Debugging of the System. The vibration char-acteristics of the linear-motor-driven water piston pump areaffected by several factors, such as mechanical structure andservo control system. Comparedwith the single controlmodeof traditional piston pump, the servo control system of thelinear-motor-driven water piston pump is able to perform avariety of servo controlmodes.The spline (i.e., spline interpo-lation algorithm) and PVT (i.e., location-time interpolationalgorithm) motion mode are the most widely used motionmodes of the linear motor. In the spline motion mode, themovement distance is divided into equal segments by the timeand only needs to define the position of the coordinate pointsat the time of operation.The advantage of this method is thatit can reduce the workload and facilitate the calculating. ThePVTmotion mode needs to define the distance, velocity, andmove time at the end of each step. Therefore, it has higherrequest for computing performance of the servo controlsystem in comparison with the spline motion mode.

Since the piston is directly connected with the linearmotor, the instantaneous flow rate of the system is propor-tional to the piston instantaneous effective velocity. Thus, theproperty parameters of linear motor, such as the followingerrors, will eventually affect the performance of linear-motor-driven water piston pump. In order to make the pump getsteady flow output and achieve the goal of low vibration andnoise, it should set the servo control characteristic parametersof linearmotor appropriately.The influencing factors of servocontrol characteristics include not only the motion modebut also the movement curve. Usually the motion curves aretriangular wave and S wave curve.

The velocity mathematic model of triangle wave planningin the time domain of [0, 𝑇] is described as follows:

𝑔 (𝑡) =

{{{{{{{

{{{{{{{

{

4𝐴

𝑇𝑡 𝑡 ≤

𝑇

4

−4𝐴

𝑇𝑡 + 2𝐴

𝑇

4< 𝑡 ≤

3𝑇

4

4𝐴

𝑇𝑡 − 4𝐴

3𝑇

4< 𝑡 ≤ 𝑇,

(22)

where 𝐴 is the speed amplitude of linear motor.

Table 2: The following errors under different modes and curves.

Maximum followingerrors (cts)

Average followingerrors (cts)

PVT/triangle curve 148.1 51.0PVT/S curve 155.3 54.6Spline/triangle curve 149.6 51.5Spline/S curve 150.0 56.3Note: 1 cts = 0.5𝜇m.

The velocity mathematic model of S wave planning in thetime domain of [0, 𝑇/2] is given by

𝑔 (𝑡)

=

{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{

{

1

2𝐽𝑡2

0 ≤ 𝑡 ≤ 𝑇1

V1+ 𝑎 (𝑡 − 𝑇

1) 𝑇

1≤ 𝑡 ≤ 𝑇

2

V2+ 𝑎 (𝑡 − 𝑇

2) −

1

2𝐽 (𝑡 − 𝑇

2)2

𝑇2≤ 𝑡 ≤ 𝑇

3

V3−

1

2𝐽 (𝑡 − 𝑇

3)2

𝑇3≤ 𝑡 ≤ 𝑇

4

V4− 𝑎 (𝑡 − 𝑇

4) 𝑇

4≤ 𝑡 ≤ 𝑇

5

V5− 𝑎 (𝑡 − 𝑇

5) +

1

2𝐽 (𝑡 − 𝑇

5)2

𝑇5≤ 𝑡 ≤ 𝑇

6,

(23)

where 𝑎 and 𝐽 are the constants and V1, V2, V3, V4, and V

5are

the speeds of each interval, respectively.Once the accelerationtime, uniformmotion time, one-waymovement distance, andmotion period are determined, the wholemotion process canbe derived. For the working condition of the linear-motor-driven water piston pump, when the four-group linear motormove at the phase difference of 90 degrees, respectively, itcould reach the constant flow output theoretically.

For the sake of obtaining the best running state, thedynamic response characteristics of the linear motor underdifferent motion modes and motion curves have been con-ducted in the simulation system. The debugging results ofthe linear motor in simulation mode under different motionmodes and curves are shown in Figure 7.

In the coordinate system, the horizontal axis representstime and the unit is the second, while the longitudinal axesrepresent the velocity and following error, and the measuringunits are cts/s and cts (cts is the counting unit of grating,and it depends on the resolution of grating; here one cts isequal to 0.5 𝜇m), respectively. From the debugging results (asshown in Table 2), it can be seen that the simulation resultsare similar and canmeet the normal operation under differentconditions.The better working condition is the triangle curveunder PVT motion mode. The average following error isclose to 51.0 cts and the maximum following error is 148.1 cts.The following error of linear motor under triangle curve islesser than that of S curve. Obviously, appropriate motionmode and curve could effectively reduce the following errorto control the actual motion track precisely and then to lowerthe vibration of the pump.


325

275

225

175

125

75

25

−25

−75

−125

−175

−225

−275

−325

Velo

city

(CTS

/s)

0 0.28 0.57 0.85 1.13 1.41 1.7 1.98 2.26 2.54

91

71

51

31

11

−9

−29

−49

−69

−89

−109

−129

−149

−169

Follo

win

g er

rors

(CTS

)Mtr 1 Act Vel (left axis)Mtr 1 Act Pos (left axis)Mtr 1 Fol Err (right axis)

×104

Time (s)

(a) Triangle velocity curves of linear motor at PVT mode

Velo

city

(CTS

/s)

Follo

win

g er

rors

(CTS

)

86

66

46

26

−14

−34

−54

−74

−94

−114

−134

−154

−174

106

×104

324

274

224

174

124

74

24

−26

−76

−126

−176

−226

−276

−326

6

0 0.29 0.58 0.87 1.16 1.45 1.74 2.03 2.32 2.61 2.9

Mtr 1 Act Vel (left axis)Mtr 1 Act Pos (left axis)Mtr 1 Fol Err (right axis)

Time (s)

(b) S velocity curves of linear motor at PVT mode

276

226

176

126

76

26

−24

−74

−124

−174

−224

−274

−324

Velo

city

(CTS

/s)

Time (s)0 0.28 0.57 0.85 1.13 1.41 1.7 1.98 2.26 2.54

91

71

51

31

11

−9

−29

−49

−69

−89

−109

−129

−149

−169

Follo

win

g er

rors

(CTS

)

×104


(c) Triangle velocity curves of linear motor at spline mode

275

225

175

125

75

25

−25

−75

−125

−175

−225

−275

−325

Velo

city

(CTS

/s)

Follo

win

g er

rors

(CTS

)

×104

0 0.28 0.57 0.85 1.13 1.41 1.7 1.98 2.26 2.55 2.83


99

79

59

39

19

−1

−21

−41

−61

−81

−101

−121

−141

Time (s)

(d) S velocity curves of linear motor at spline mode

Figure 7: Debugging diagram under different motion modes and curves.

3.2. Vibration Experiment. Thevibration source of the linear-motor-driven water piston pump mainly comes from themotion of mechanism and the flow pulsation noise. Themain effect factors of the vibration response characteristicsin this pump are the response lag of distributing valve andsynchronous phase error of linear motor. Therefore, underthe certain system structure parameters, the control strategyof linear motor has become the key factor for influencing the

system vibration characteristics. On the basis of the abovesimulation analysis, the vibration experiment is conducted toobtain the vibration and noise information.

As shown in Figure 8, the cylinder wall is chosen as thetest object. The three channels gather the vibration signalof x-axis (the motion direction of linear motor), y-axis(perpendicular to the linear motor motion direction in thehorizontal plane), and z-axis (vertical direction), respectively.


Table 3: Vibration amplitude of each channel under the trianglewave motion curve.

Signal channel Maximum amplitude(𝜇m)

Minimum amplitude(𝜇m)

1 413.2 −454.02 2673.6 −2972.23 742.4 −655.8

Table 4: Vibration amplitude of each channel under the S wavemotion curve.

Signal channel Maximum amplitude(𝜇m)

Minimum amplitude(𝜇m)

1 634.6 −423.72 6706.8 −7312.93 903.5 −1071.4

Figure 8: Vibration test point.

Though Coinv DASP software, the acceleration, velocity,and displacement of the vibration signal are collected in realtime. According to previous dynamic characteristic simula-tion analysis of the linear motor, the vibration response testof triangle curve and S curve under PVT motion mode havebeen conducted.Themotion period is set as 0.6 seconds, andthe stroke is 205mm. Four groups of linearmotor aremovingat the phase difference of 90 degrees. After the calculusprocessing of the collected vibration data, the amplitudedistribution in the time domain of the x-, y-, and z-axis isshown in Figure 9.

It can be seen from Figure 9 that the periodic vibrationexists in the system and the vibration impact cycle is roughlyidentical to the system movement cycle. It demonstratesthat the main vibration source of the linear-motor-drivenwater piston pump is continuous high-speed reciprocatingmotion of the linear motor unit. The vibration impact isderived from the unbalanced torque caused by the thrust ofthe linear motor. The vibration amplitudes of each channelunder the S and triangle curve are shown in Tables 3 and 4,respectively. Compared to the results of three signal channels,the vibration peak appears in number 2 signal channeland the maximum amplitude of triangle and S curve are2673.6 𝜇m and 6706.8𝜇m, respectively. It means that severevibration occurs in the direction which is perpendicular to

the linear motor movement direction.The dampers mountedat the platform bottom and cylinder wall have a good effectto reduce the vibration. From another aspect, the base witha certain degree of flexibility will aggravate the vibration ofsystem.Therefore, the vibration amplitude of number 3 signalchannel is greater than that of number 1.

From the previous parameters debugging of simulationsystem, it can be known that the following errors of linearmotor under triangle curve are lesser than that of S curve.The following error was ultimately reflected in the vibrationamplitude of test results. Comparing the vibration amplitudes(as shown in Tables 3 and 4), the vibration of the triangularwave is significantly better than that of S wave, and this isconsistent with the simulation results. By adopting triangularmotion curve under PVT mode, the cylinder vibrationamplitude was reduced 35%, 60%, and 18% in x-, y-, and z-axis direction, respectively.

3.3. Power Spectrum Analysis. Random vibration is the mostcommon type of vibration, and it should be studiedwhen car-rying out a vibration-proof design. Power spectrum density(PSD) is an effective method to describe random vibration.Generally, the randomvibration is characterized by the powerspectral density function, for a time series 𝜒(𝑡) is defined asthe Fourier transform of the correlation function 𝑅

𝜒(𝜏) [24]:

𝐺𝜒𝑦

(𝜔) =1

2𝜋∫

∞

−∞

𝑅𝜒𝑦

(𝜏) 𝑒−𝑖𝜔𝜏

𝑑𝜏. (24)

The correlation function is used to describe the relationshipbetween the two values of the same signal in the interval 𝜏and can be written as

𝑅 (𝜏) = lim𝑇→∞

1

𝑇∫

𝑇

0

𝑥 (𝑡) 𝑥 (𝑡 + 𝜏) 𝑑𝑡, (25)

where 𝑥(𝑡) is the vibration response of any transient 𝑡.The PSD is widely used because the modes could be

indicated clearly by spectral peaks. Power spectrum is theconcept of statistical average randomprocess that is expressedby the signal power with the change of frequency. Thehorizontal axis unit of power spectrumdensity plot is Hz, andthe vertical axis is g2/Hz,where g2 represents themean squarevalue of the acceleration. With the DASP software to analyzethe collected data, the power spectrum of the linear-motor-driven water piston pump under different motion conditionis shown in Figure 10 and the power spectrums of each signalchannel are displayed in different graph.

Compared to the graphs of signal channel in the differentmotion mode, it can be observed that the trend of powerspectrum is basically identical. Like the previous analysis,the power spectrum only reserves the amplitude informationand lost the phase information. So, the difference lies in theamplitude at the resonance frequency.The amplitudes of eachchannel under triangular curve are 0.025, 0.082, and 0.078 g2,while the amplitude of each channel under S curve are 0.16,0.31, and 0.36 g2. Obviously, the vibration amplitude of the Swave is 4∼6 times that of triangular wave. The experimentaldata show that the vibration under triangular motion curve


200

0

−2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2000

−200

(1D)

(s)

1000

0

−1000

(2D)

(3D)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(s)

(𝜇m

)(𝜇

m)

(𝜇m

)

(a) Triangle curve, PVT motion mode

600200

−200−600

80004000

0−4000

−1000

1000

−8000

0

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(s)

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(s)

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

(s)

(1D)

(2D)

(3D)

(𝜇m

)(𝜇

m)

(𝜇m

)

(b) S curve, PVT motion mode

Figure 9: Vibration amplitude distribution in the time domain of the x-, y-, and z-axis.

0 500 1k 1500 2k 2500

(Hz)

(1)

0 500 1k 1500 2k 2500

(Hz)

(3)

0.05

0.04

0.03

0.02

0.01

0.00

0.10

0.08

0.06

0.04

0.02

0.00

0.10

0.08

0.06

0.04

0.02

0.00

500 1k 1500 2k 2500

(Hz)

(2)

(g2)

(g2)

(g2)

(a) Triangle curve, PVT motion mode

0.5

0.4

0.3

0.2

0.1

0.00 500 1k 1500 2k 2500

(Hz)

0.5

0.4

0.3

0.2

0.1

0.0

0.5

0.4

0.3

0.2

0.1

0.0

(1)

0 500 1k 1500 2k 2500

(Hz)

(3)

500 1k 1500 2k 2500

(Hz)

(2)

(g2)

(g2)

(g2)

(b) S curve, PVT motion mode

Figure 10: Power spectrum of linear-motor-driven water piston pump system under triangle and S motion curve.


is much better than that of S curve, which is consistent withthe previous simulation conclusion. Different motion curveswill affect the following errors of the linear motor, which isreflected in the vibration of the system finally.

It can be seen from Figure 10(a) that, under the triangularwave motion curve, the peak of number 1 signal channel isonly one-third the peaks of number 2 and number 3.Thepeakfrequency of number 1 signal channel (the motion directionof linear motor) only appears near 500Hz and has a little dif-ferencewith the value provided by the FEManalysis.With thedampers equipped on the cylinder body in the motion direc-tion of linear motor, it will inevitably lead to the differencebetween the simulation results and the experimental data.According to the data collected by number 2 signal chan-nel, the resonance frequency is mainly concentrated in thevicinity of 500Hz and 800Hz. Similarly, the analysis resultof number 3 signal channel is essentially consistent with theharmonic analysis result. Therefore, the resonance frequencyof linear-motor-driven water piston pump is concentratedupon 500Hz and 800Hz in the low frequency range. Thedampers can change the resonance frequency of the systemto a certain extent. During the operation of the linear-motor-driven water piston pump, the resonance frequency rangedetected by simulation analysis and experiment test shouldbe avoided to effectively reduce the vibration.

4. Conclusion

Based on the prototype and three-dimensional model ofthe linear-motor-driven water piston pump, this paper hascarried out the vibration characteristic analysis. The finiteelement analysis software has been employed to perform astatic stress, modal, and harmonic analysis. The maximumdeformation and equivalent stress of the mechanical struc-ture were measured at the rated working condition. Fromthe results presented in this paper, the important conclusionsobtained can be drawn as follows:

(1)Thesimulation results reveal that themaximumdefor-mation and equivalent stress of the mechanical structure inthis pump are 0.15mm and 95.398MPa, respectively, whichis lower than the yield strength of the material and theallowed deformation. The first-order natural frequency isclose to 166.19Hz, which is much larger than the systemmotion frequency (3Hz), and themodal deformation is about18.3mm. And the mode shape under low frequency stage isassociated with the eccentricity swing of the piston rod andthe maximum deformation is within the acceptable range.

(2) The harmonic analysis and vibration experimentresults show that the resonance frequency of linear-motor-driven water piston pump is concentrated upon 500Hz and800Hz in the low frequency range. The dampers can changethe resonance frequency of the system to a certain extent.

(3)The vibration under triangular motion curve is muchbetter than that of S curve, which is consistent with theprevious simulation conclusion.

The research provides an effective method to detect thevibration characteristics of the linear-motor-driven waterpiston pump and also a reference for design and optimizationof the piston pump.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank the National Natural ScienceFoundations of China (nos. 51375018 and 11572012), NationalHigh-Tech R&D (863) Program (no. 2012AA091103), BeijingNatural Science Foundation (no. 3164039), and China Post-doctoral Science Foundation (no. 2015M580946) for theirfunding for this research.

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