research article vibration modes and the dynamic behaviour...

Research ArticleVibration Modes and the Dynamic Behaviour ofa Hydraulic Plunger Pump

Tianxiao Zhang and Nong Zhang

School of Electrical,Mechanical andMechatronic Systems, University of Technology Sydney, 15 Broadway, Ultimo, NSW2007, Australia

Correspondence should be addressed to Tianxiao Zhang; [email protected]

Received 19 August 2015; Revised 19 November 2015; Accepted 22 November 2015

Academic Editor: Vadim V. Silberschmidt

Copyright © 2016 T. Zhang and N. Zhang. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Mechanical vibrations and flow fluctuation give rise to complex interactive vibrationmechanisms in hydraulic pumps.Theworkingconditions for a hydraulic pump are therefore required to be improved in the design stage or as early as possible. Considering thestructural features, parameters, and operating environment of a hydraulic plunger pump, the vibration modes for two-degree-of-freedom systemwere established by using vibration theory and hydraulic technology. Afterwards, the analytical form of the naturalfrequency and the numerical solution of the steady-state response were deduced for a hydraulic plunger pump.Then, a method forthe vibration analysis of a hydraulic pump was proposed. Finally, the dynamic responses of a hydraulic plunger pump are obtainedthrough numerical simulation.

1. Introduction

Since there have been increasingly higher requirementsimposed on the engineering quality, the accuracy, and reli-ability of products, it is an urgent task to study and solvevarious vibration problems existing in industrial machinery.Due to increasing system complexity, rapid operations, andimproved accuracy of mechanical devices, vibration is aserious issue. Therefore, both the static strength effect andthe dynamic force effect must be taken into account whiledesigning mechanical devices [1–3].

Many studies have been performed to analyse thedynamic performance of hydraulic components and systems.To prevent hydraulic systems from breaking down, analysisof their vibration is required. An effective method of reduc-ing vibration and noise needs to be developed: its aim isto improve the performance of hydraulic devices and thusreduce the vibration and noise in hydraulic systems. Vibra-tion and noise arise from the interaction between solidsand the fluid here. Fluid-structure interaction (FSI) not onlyreflects the essence of vibration noise in hydraulic systembut also is the leitmotif for research in the field. Fluid-struc-ture interaction would be caused between solid and liquidunder different conditions. The field of hydraulic model and

vibration has been discussed in some research [4–7]. A blockdiagram for the vibration analysis of a hydraulic componentand system is shown in Figure 1.

A hydraulic pump is the main vibration and noise sourcein a hydraulic system, and its working state determines thesafe operation of hydraulic components therein. Therefore, itcan be seen that mechanical vibrations and flow fluctuationsnot only affect the engineering quality but also reduce thelifespan of hydraulic components and systems, generatenoise pollution, and even cause damage. Furthermore, theymay give rise to accidents. Studies relating the vibrationof hydraulic pump mainly focus on the analysis of testdata and the reduction in vibration and noise [8–16], andquantitative simulations have also been conducted to analysethe vibrations thereof. The vibration analysis of a hydraulicpump is beneficial for controlling noise and vibration.

Influenced by the design, structure, and operating envi-ronment of a hydraulic plunger pump as well as the inherentcharacteristic curve, flow pulsation is bound to be generated.Flow and pressure pulsation are two of the leading reasonsfor noise and vibration being generated by a hydraulicpump. This paper gave full considerations to the vibrationproblems caused by flow and pressure pulsation in a hydraulicplunger pump and converted the simplified formula into an

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 9679542, 7 pageshttp://dx.doi.org/10.1155/2016/9679542

2 Shock and Vibration

Vibration analysis on the hydraulic component and system

Vibration theory of solidVibration theory of fluidVibration theory of FSI

Vibrationtheory

Hydraulic technology (i) Avoid the

breakdown or failure of hydraulic system and device(ii) Reduce the vibration noise pollution of hydraulic system and device

(iii) Improve the performance ofhydraulic system and device

High speed

High power

High pressure

Precise treatment

Light weight

Intellectualization

High reliability

Humanization

Environment-friendly

+

Figure 1: Vibration analysis: hydraulic component and system.

F2(t) x2(t) F1(t) x1(t)

k3

c3

m2

c2

k2

m1

k1

c1

Figure 2: Vibration model for the two degrees of freedom of ahydraulic plunger pump.

excitation describing the vibration system of a plunger pump.In addition, a vibration mode for the two degrees of freedomin a hydraulic plunger pump was made available for practicalcalculation and was established to estimate the vibration of ahydraulic plunger pump under specific working conditions.Meanwhile, themodel reveals the basic mechanisms of vibra-tion and noise in a hydraulic plunger pump. By conductingthe dynamic analysis using the proposed approach, a betterunderstanding of hydraulic plunger pumps can be obtained.

2. Vibration Model forTwo-Degree-of-Freedom System fora Hydraulic Plunger Pump

2.1. Vibration Model of Plunger Pump. The vibration modelfor the two degrees of freedom of a hydraulic plunger pumpwas developed based on the data and conditions regardingthe structural features, parameters, variables, constraints,operational states, and flow pulsation. The model is shownin Figure 2. It was assumed that the mass of the cylinderblock of the hydraulic plunger pump was 𝑚

1(kg), and the

masses of its retainer plate, seven plungers, and sliding shoeswere merged and represented by 𝑚

2(kg). The connection

stiffness and the damping between the cylinder block andport plate were, respectively, set to 𝑘

1(N/m) and 𝑐

1(N⋅s/m).

The cylinder and retainer plate were connected by a centralspring whose stiffness and damping were 𝑘

2(N/m) and 𝑐

2

(N⋅s/m), respectively. The sliding shoes were connected withan angle plate with a connection stiffness and damping of 𝑘

3

(N/m) and 𝑐3(N⋅s/m), respectively.

According to the proposed vibration model, the positivedirections of the acceleration and excitationwere determined,which were in accordance with the positive direction of thecoordinate axes. The deduced vibration differential equationis

[

𝑚1

0

0 𝑚2

]{

�̈�1

�̈�2

} + [

𝑐1+ 𝑐2

−𝑐2

−𝑐2

𝑐2+ 𝑐3

]{

�̇�1

�̇�2

}

+ [

𝑘1+ 𝑘2

−𝑘2

−𝑘2

𝑘2+ 𝑘3

]{

𝑥1

𝑥2

} = {

𝐹1(𝑡)

−𝐹2(𝑡)

} ,

(1)

where the constant matrices includingM = [𝑚], C = [𝑐], andK = [𝑘], which are constituted by coefficient matrices, are,respectively, the mass matrix, damping matrix, and stiffnessmatrix: the vector x = {𝑥} is the displacement vector.

2.2. Flow Pulsation. One of the most influential factors inthe vibration model of hydraulic plunger pump is the valueof input vibration excitation; because flow pulsation is thesource of pressure pulsation, flow pulsation has to be ana-lysed to study pressure pulsation. Flow pulsation refers tothe instantaneous flow variation when a hydraulic pump isrunning. When a hydraulic pump keeps operating contin-uously, the ever-changing sealing volume is expected to begenerated in a majority of hydraulic pumps. Meanwhile, theinstantaneous flow changes repeatedly, and some instanta-neous, nonconstant flow may generate flow pulsation. Theinstantaneous actual flow of a hydraulic plunger pump maybe given as follows.

When 0 ≤ 𝜑 ≤ 𝛼 and 𝑧0= (𝑧 + 1)/2,

𝑞𝑠1= 𝐴𝑅𝜔𝜂

𝑉tan 𝛾

cos (𝛼/2 − 𝜑)2 sin (𝛼/2)

=

𝜋𝑑2𝑅𝜔𝜂𝑉tan 𝛾

8 sin (𝛼/2)cos (𝛼/2 − 𝜑) .

(2a)

When 𝛼 ≤ 𝜑 ≤ 2𝛼 and 𝑧0= (𝑧 − 1)/2,

𝑞𝑠2= 𝐴𝑅𝜔𝜂

𝑉tan 𝛾

cos (3𝛼/2 − 𝜑)2 sin (𝛼/2)

=


8 sin (𝛼/2)cos (3𝛼/2 − 𝜑) ,

(2b)

Shock and Vibration 3

where 𝐴 represents the cross-sectional area (m2) of theplunger;𝑅 is the radius (m) of the central circle of the plunger,and 𝑑 indicates the diameter (m) of the plunger; 𝛾 denotesthe inclination angle (rad) of the angle plate, 𝑧 is the numberof the plungers, and 𝜔 is the angular velocity (rad/s) of theoil cylinder block; 𝑛 is the rotational speed (rpm) of theplunger pump, 𝜂

𝑉is the volumetric efficiency, and 𝛼 is the

half included angle (rad) between two adjacent plungers and𝛼 = 𝜋/𝑧; 𝜑 represents the angular displacement (rad) of thecylinder block, and 𝜑 = 𝜔𝑡. Let

𝐾𝑞=


8 sin (𝛼/2)=


8 sin (𝜋/2𝑧). (3)

Then

𝑞𝑠1= 𝐾𝑞cos(𝛼

2

− 𝜑) , (4a)

𝑞𝑠2= 𝐾𝑞cos(3𝛼

2

− 𝜑) . (4b)

Converting the instantaneous actual flow into a functionof time 𝑡(𝑠), then

𝑞𝑠1= 𝐾𝑞cos(𝜔𝑡 − 𝜋

2𝑧

) in case 0 ≤ 𝑡 ≤

𝜋

𝑧𝜔

, (5a)

𝑞𝑠2= 𝐾𝑞cos(𝜔𝑡 − 3𝜋

2𝑧

) in case 𝜋

𝑧𝜔

≤ 𝑡 ≤

2𝜋

𝑧𝜔

, (5b)

where 𝑞𝑠1and 𝑞

𝑠2represent the instantaneous flow through

the plunger pump. It can be seen that when 𝜑 = 𝛼/2 and 𝜑 =

3𝛼/2 (namely, 𝜑 = 𝜋/2𝑧 and 𝜑 = 3𝜋/2𝑧), instantaneous flowwas at amaximum, 𝑞

𝑠max; when𝜑 = 0 and𝜑 = 𝛼 (namely,𝜑 =

0 and 𝜑 = 𝜋/𝑧), instantaneous flow was at a minimum, 𝑞𝑠min.

Therefore, the flow pulsation coefficient 𝛿𝑞can be defined as

follows:

𝛿𝑞=

𝑞𝑠max − 𝑞𝑠min

𝑞𝑡

, (6)

where 𝑞𝑡is the theoretical mean flow. When the cylinder

block of the plunger pump rotates through one completerevolution, each plungermoves back and forth once in a cycleof oil adsorption and extrusion. Therefore, the theoreticalmean flow 𝑞

𝑡and the actual flow 𝑞 can be presented as

𝑞𝑡= 2𝐴𝑧𝑛𝑅 tan 𝛾 = 15𝑧𝜔𝑑

2𝑅 tan 𝛾,

𝑞 = 2𝐴𝑧𝑛𝑅𝜂𝑉tan 𝛾 = 15𝑧𝜔𝑑

2𝑅𝜂𝑉tan 𝛾.

(7)

2.3. Pressure Pulsation. Flow pulsation inevitably gives rise topressure pulsation, which indicates that the flow and pressureoutput from the hydraulic plunger pump change with time.Therefore, the flow and pressure output by hydraulic plungerpump are not necessarily stable. The change in the volumeof a hydraulic plunger pump always results in fluctuations inoutput pressure and fluid flow, leading to the generation ofnoise and vibration. According to the fundamental principlesof fluid dynamics, the change of the flow in the closed cavitybetween the plunger and cylinder block of a hydraulic plunger

pump inevitably leads to a pressure change. With respect toa compressible flow, its instantaneous pressure can be repre-sented by the following formulas.

When 0 ≤ 𝑡 ≤ 𝜋/𝑧𝜔,

𝑝𝑠1=

𝐸𝑞

7𝑉𝑞

∫

𝑡

0

𝑞𝑠1d𝑡 =

𝐸𝑞𝐾𝑞

𝑉𝑞

∫

𝑡

0

cos(𝜔𝑡 − 𝜋

2𝑧

) d𝑡

=

𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔

[sin(𝜔𝑡 − 𝜋

2𝑧

) + sin( 𝜋

2𝑧

)] .

(8a)

When 𝜋/𝑧𝜔 ≤ 𝑡 ≤ 2𝜋/𝑧𝜔,

𝑝𝑠2=

𝐸𝑞

7𝑉𝑞

∫

𝑡

𝜋/𝑧𝜔

𝑞𝑠2d𝑡

=

𝐸𝑞𝐾𝑞

7𝑉𝑞

[∫

𝑡

𝜋/𝑧𝜔

cos(𝜔𝑡 − 3𝜋

2𝑧

) d𝑡]

=

𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔

[sin(𝜔𝑡 − 3𝜋

2𝑧

) + sin( 𝜋

2𝑧

)] ,

(8b)

where 𝐸𝑞represents the elastic modulus of the flow (Pa) and

the volume of the closed cavity for a single plunger is 𝑉𝑞=

𝐴𝑑𝑠 = (𝜋𝑑

2/2)𝑅 tan 𝛾.

2.4. Fourier Series Simulation. The flow and pressure pulsa-tion in a hydraulic plunger pump are a nonharmonic periodicfunction, which can be represented by a harmonic Fourierseries. Thereby, the dynamic response problem correspond-ing to the harmonic excitation of its Fourier series can besolved. The periodic excitation function in an interval of[𝑒1, 𝑒2] is unfolded as the Fourier series; namely,

𝐹 (𝑥) =

𝑎0

2

+

∞

∑

𝑗=1

(𝑎𝑗cos

𝑗𝜋 (2𝑥 − 𝑒1− 𝑒2)

𝑒2− 𝑒1

+ 𝑏𝑗sin

𝑗𝜋 (2𝑥 − 𝑒1− 𝑒2)

𝑒2− 𝑒1

) =

𝑎0

2

+

∞

∑

𝑗=1

[𝑎𝑗cos 𝑗 (𝑧𝜔𝑥 − 𝜋) + 𝑏

𝑗sin 𝑗 (𝑧𝜔𝑥 − 𝜋)] ,

(9)

where 𝑒1= 0 and 𝑒

2= 2𝜋/𝑧𝜔. Equation (9) shows that

a complex periodic excitation function can be decomposedinto the superposition of a series of harmonic functions. Thecoefficients of Fourier series 𝑎

0, 𝑎𝑗, and 𝑏

𝑗can be determined

as follows:

𝑎𝑗=

2

𝑒2− 𝑒1

∫

𝑒2

𝑒1

𝐹 (𝑡) cos𝑗𝜋 (2𝑡 − 𝑒

1− 𝑒2)

𝑒2− 𝑒1

d𝑡

=

𝑧𝜔

𝜋

∫

2𝜋/𝑧𝜔

0

𝐹 (𝑡) cos 𝑗 (𝑧𝜔𝑡 − 𝜋) d𝑡

(𝑗 = 0, 1, 2, 3, . . .) ,

(10a)


𝑏𝑗=

2

𝑒2− 𝑒1

∫

𝑒2

𝑒1

𝐹 (𝑡) sin𝑗𝜋 (2𝑡 − 𝑒

1− 𝑒2)

𝑒2− 𝑒1

d𝑡

=

𝑧𝜔

𝜋

∫

2𝜋/𝑧𝜔

0

𝐹 (𝑡) sin 𝑗 (𝑧𝜔𝑡 − 𝜋) d𝑡

(𝑗 = 1, 2, 3, . . .) .

(10b)

The function 𝐹(𝑥) can be represented by Fourier seriesas long as the defined integrations of 𝑎

𝑗and 𝑏𝑗actually exist.

If 𝐹(𝑥) cannot be represented by means of function, it canbe calculated through approximate calculation. The periodicpressure pulsation indicated by formulas (8a) and (8b) isunfolded as the Fourier series, and the coefficient of Fourierseries is

𝑎0=

2𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔2sin( 𝜋

2𝑧

) , (11a)

𝑎𝑗=

𝑧𝜔

𝜋

𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔

[∫

𝜋/𝑧𝜔

0


2𝑧

) + sin( 𝜋

2𝑧

)]

⋅ cos 𝑗 (𝑧𝜔𝑡 − 𝜋) d𝑡] + 𝑧𝜔

𝜋

𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔

⋅ ∫

2𝜋/𝑧𝜔

𝜋/𝑧𝜔


2𝑧

) + sin( 𝜋

2𝑧

)]

⋅ cos 𝑗 (𝑧𝜔𝑡 − 𝜋) d𝑡 = 0 (𝑗 = 1, 2, 3, . . .) ,

(11b)

𝑏𝑗=

𝑧𝜔

𝜋

𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔

∫

𝜋/𝑧𝜔

0


2𝑧

) + sin( 𝜋

2𝑧

)]

⋅ sin 𝑗 (𝑧𝜔𝑡 − 𝜋) d𝑡 + 𝑧𝜔

𝜋

𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔

⋅ ∫

2𝜋/𝑧𝜔

𝜋/𝑧𝜔


2𝑧

) + sin( 𝜋

2𝑧

)]

⋅ sin 𝑗 (𝑧𝜔𝑡 − 𝜋) d𝑡 = 𝑧𝜔

𝜋

𝐸𝑞𝐾𝑞

7𝑉𝑞𝜔

{−

1

(1 + 𝑗𝑧) 𝜔

⋅ sin( 𝜋

2𝑧

+ 𝑗𝜋) +

1

(1 − 𝑗𝑧) 𝜔

sin( 𝜋

2𝑧

− 𝑗𝜋)}

(𝑗 = 1, 2, 3, . . .) .

(11c)

Therefore, the quantity of pressure pulsation induced bythe quantity of flow pulsation can be expressed as

𝑝𝑠(𝑡) =

𝑎0

2

+

∞

∑

𝑗=1

𝑏𝑗sin 𝑗 (𝑧𝜔𝑡 − 𝜋) (𝑗 = 1, 2, 3, . . .) , (12)

where 𝑝𝑠is the quantity of pressure pulsation of the plunger

pump. The sum of pressure pulsation and mean pressureexpresses the pressure work in pump; that is,𝑝(𝑡) = 𝑝

0+𝑝𝑠(𝑡).

That means the pressure work in pump is always greater thanzero. According to (12), the multiseries pressure pulsationfunction can be simulated using a Fourier series, and Figure 3shows the pressure pulsation function as 𝑗 (𝑗 = 1, 2, . . . , 10)with its different values.

From Figure 3, the quantity of pressure pulsation changereciprocated at mean pressure port, positive value of verticalaxis expresses the direction of pulsation quantity which is thesame as 𝑥-axis, negative value of vertical axis expresses thedirection of pulsation quantity which is opposite to 𝑥-axis,and the pulsation quantity function is periodical change invalue and direction. Therefore, as long as relevant conditionsare satisfied, all periodic functions can be described as har-monic, according to Fourier series; the complicated quantityof pressure pulsation function is expressed as harmonicfunction, convergent Fourier series: this solves the responseproblem of harmonic excitation and caused piston pump tomove circularly.

2.5. System Excitation. The excitation of the vibration systemfor the two-degree-of-freedom system of a hydraulic plungerpump caused by flow and pressure pulsation is given by

𝐹1(𝑡) = 𝑝

𝑠𝐴𝐶, (13a)

where 𝐴𝐶represents the cross-sectional area of the cylinder

hole of the cylinder block, 𝐴𝐶

= 7(𝜋/4)𝑑2, and 𝑑 is the

diameter of piston. Consider

𝐹2(𝑡) = 7𝑝

𝑠𝐴𝑃, (13b)

where 𝐴𝑃is the cross-sectional area of a single plunger and

here 𝐴𝐶≈ 7𝐴𝑃; therefore, the excitation 𝐹

2(𝑡) is considered

to be equal to−𝐹1(𝑡), where𝐹

1(𝑡) and𝐹

2(𝑡) are a pair of action

and counteraction forces with the samemagnitude but actingin opposite directions.

3. Vibration of the Two-Degree-of-FreedomSystem of a Hydraulic Plunger Pump

3.1. The Natural Frequency and Modes of Vibration of theSystem. As seen from (1), matrices M = [𝑚], C = [𝑐],and K = [𝑘] are symmetrical, and the matrix elements are,respectively, listed as follows:

𝑚11= 𝑚1,

𝑚12= 𝑚21= 0,

𝑚22= 𝑚2,

𝑐11= 𝑐1+ 𝑐2,

𝑐12= 𝑐21= −𝑐2,

𝑐22= 𝑐2+ 𝑐3,

𝑘11= 𝑘1+ 𝑘2,

𝑘12= 𝑘21= −𝑘2,

𝑘22= 𝑘2+ 𝑘3.

(14)


j = 1

j = 2

j = 3

j = 5

j = 10

×107

−5

−4

−3

−2

−1

0

1

2

3

4

5

0.005 0.01 0.015 0.02 0.025 0.030t (s)

ps

(Pa)

Figure 3: Pressure pulsation 𝑝𝑠(Pa).

To study the inherent characteristics of a hydraulicplunger pump, the deduced characteristic equation, or fre-quency equation, of the two-degree-of-freedom system of ahydraulic plunger pump may be given as follows:

Δ (𝜔2) = 𝑚

1𝑚2𝜔4− (𝑚1𝑘22+ 𝑚2𝑘11) 𝜔2+ 𝑘11𝑘22

− 𝑘2

12= 0.

(15)

Equation (15) is a quadratic equation in 𝜔2, and its rootsare

𝜔2

1

𝜔2

2

=

1

2

𝑚1𝑘22+ 𝑚2𝑘11

𝑚1𝑚2

∓

1

2

√(

𝑚1𝑘22+ 𝑚2𝑘11

𝑚1𝑚2

)

2

− 4

𝑘11𝑘22− 𝑘2

12

𝑚1𝑚2

.

(16)

Since 𝑘11𝑘22

= (𝑘1+ 𝑘2)(𝑘2+ 𝑘3) > 𝑘

2

2= 𝑘2

12, as

seen in (16), the value of the item following the ∓ sign wassmaller than that to the left of it, and thus 𝜔2

1and 𝜔

2

2are

positive numbers. Therefore, two positive real roots of thecharacteristic equation in (15) can be obtained, namely, thetwo natural frequencies 𝜔

1and 𝜔

2of the system.

𝑢(1)

1and 𝑢

(1)

2are used to represent the amplitudes cor-

responding to 𝜔1, and 𝑢

(2)

1and 𝑢

(2)

2are the amplitudes

corresponding to 𝜔2. The homogeneous differential equation

can only determine the ratios of 𝑢(1)2/𝑢(1)

1and 𝑢

(2)

2/𝑢(2)

1. The

deduced amplitude ratios are shown as follows:

𝑟1=

𝑢(1)

2

𝑢(1)

1

= −

𝑘11− 𝜔2

1𝑚1

𝑘12

= −

𝑘12

𝑘22− 𝜔2

1𝑚2

, (17a)

𝑟2=

𝑢(2)

2

𝑢(2)

1

= −

𝑘11− 𝜔2

2𝑚1

𝑘12

= −

𝑘12

𝑘22− 𝜔2

2𝑚2

. (17b)

Therefore, the constants in pairs such as 𝑢(1)1

and 𝑢(1)2

and𝑢(2)

1and 𝑢

(2)

2can determine the natural vibration modes or

main vibration mode displayed by the system when the sys-tem conducts synchronous harmonic motion, respectively, atfrequencies of 𝜔

1and 𝜔

2. These constants can be expressed

by the following matrices:

u(1) = {

𝑢(1)

1

𝑢(1)

2

} = 𝑢(1)

1{

1

𝑟1

} , (18a)

u(2) = {

𝑢(2)

1

𝑢(2)

2

} = 𝑢(2)

1{

1

𝑟2

} . (18b)

There are two natural frequencies in this system, andcorrespondingly there are two natural vibration modes.The lower frequency 𝜔

1is the first-order natural frequency

or, simply, the fundamental frequency, while the higherfrequency 𝜔

2is the second-order natural frequency. The

corresponding vibrationmodes u(1) and u(2) are, respectively,the first-order and second-order natural vibration modes.

3.2. Dynamic Response of the Pump System. With the im-provement and development of computer software, hard-ware, and technology, almost all engineering problems canbe simulated quantitatively with high precision. Numericalsimulation involves the solution of a mathematical problemwhose exact solution is hard to find in practical engineering.The numerical simulation applied to mechanical vibrationproblems must discrete the time history for a dynamicresponse within the time domain so as to discretize thedifferential equation of motion and numerical equations atdifferent moments. Meanwhile, the speed and accelerationat a given time are described by the combination of thedisplacements at adjacent time steps. As a result, the differ-ential equation of motion of the system is converted intoalgebraic equations at discrete time steps. Then, the valuescorresponding to a series of discrete times are obtained bynumerical integration of the differential equation of motionof the coupling system. There are many commonly usedapproaches for finding the dynamic response of such systems,including (1) central difference, (2) Houbolt, (3) Wilson-𝜃,and (4) Newmark-𝛽.

4. A Numerical Example

Themass of the cylinder block of a certain hydraulic plungerpump was 𝑚

1= 2.7 kg, and the masses of the retainer plate,

seven plungers, and its sliding shoes were merged togetheras 𝑚2= 1.8825 kg. Suppose that the connection stiffness

and damping between the cylinder block and the port platewere, respectively, 𝑘

1= 3.6 × 10

7N/m and 𝑐1= 329.0N⋅s/m,

and the stiffness and damping between the cylinder and theretainer plate were 𝑘

2= 5.0 × 10

5N/m and 𝑐2= 223.0N⋅s/m,

respectively; the stiffness and damping between the slidingshoes and the swash plate were, respectively, 𝑘

3= 7.7 ×

107N/m and 𝑐

3= 490.0N⋅s/m. The inclination angle of the

swash plate was 𝛾 = 19.5∘, the radius of the central circle of

the plunger was 𝑅 = 0.285m, the diameter was 𝑑 = 0.019m,


and number of the plungers was 𝑧 = 7. Meanwhile, the rateof rotation of the plunger pump was 𝑛 = 1500 rpm. Theelastic modulus of the flow and the volumetric efficiency ofthe plunger pump were, respectively, 𝐸

𝑞= 1.226 × 10

3MPaand 𝜂𝑉= 0.95. This research attempted to obtain the natural

frequency and vibration modes of the system as well as itsvibration response.

According to the known conditions, here we have

𝑘11= 𝑘1+ 𝑘2,

𝑘12= 𝑘21= −𝑘2,

𝑘22= 𝑘2+ 𝑘3.

(19)

By substituting the above into (16), the natural frequencycould be obtained:

𝜔1= 3676

1

s,

𝜔2= 6416

1

s.

(20)

The values of 𝑟1and 𝑟2can be obtained by substituting𝜔2

1and

𝜔2

2into (17a) and (17b):

𝑟1= 0.0096,

𝑟2= −149.3203.

(21)

The natural modes of vibration for the system can becalculated according to (18a) and (18b):

u(1) = {

1

0.0096

} ,

u(2) = {

1

−149.3203

} .

(22)

Figure 4 shows the two natural modes of vibration: inthe first-order vibration mode, the two masses move forwardwith an amplitude ratio of 1 : 0.0119; in the second-ordervibration mode, the two masses move adversely with anamplitude ratio of 1 : 120.9773. It is noted that there is a nodalpoint of zero displacement in the second-order natural modeof vibration.

Newmark-𝛽 method was applied to the vibration modelhere, and the values of 𝛽 and 𝛿 were taken, respectively, as𝛽 = 1/4 and 𝛿 = 1/2. The displacement 𝑥

1(m) of mass 𝑚

1

and displacement 𝑥2(m) ofmass𝑚

2over timewere as shown

in Figure 5.As discovered during calculation, owing to the unstable

initial running stage of the hydraulic plunger pump and thesuperposition of transient and steady-state vibrations, thesystem vibrated irregularly at large amplitude. However, thetransient vibration gradually weakened and finally vanishedafter a period of time, and the system vibration reached asteady state. The amplitude of steady-state vibration of theplunger pump was acceptable in its smooth running stage.Meanwhile, comparedwith the retainer plate, seven plungers,

0.00961

1

𝜔1

𝜔2

−149.3203

u(1)

u(2)

Figure 4: Natural mode of vibration of the two-degree-of-freedomsystem.

x2

x1

t (s)0.030.0250.020.0150.010.0050

−6

−4

−2

0

2

4

6x

(m)

×10−3

Figure 5: The response of the two-degree-of-freedom system withtime 𝑡.

and sliding shoes, the amplitude of the cylinder block of theplunger pump is much greater. The numerical calculationresults can provide quantitative theoretical support for theaccurate design of a hydraulic plunger pump.

5. Conclusions

Themechanical vibration analysis and the dynamic design arekey points during the engineering design ofmechanical prod-ucts and are crucial for producing products with the requireddynamic characteristics. While conducting vibration analysisand dynamic design on a hydraulic plunger pump using theproposed approach, the actual design level and dynamic char-acteristics of the hydraulic plunger pump are expected to beimproved. This can also prevent malfunctions and accidentscaused by breakdowns. Based on using vibration theory andhydraulic technology, this research has developed a vibration


model for the dynamic analysis of a two-degree-of-freedomhydraulic plunger pump system. Meanwhile, the inher-ent characteristics and dynamic response of the hydraulicplunger pump were studied and were used to estimate thevibration of a hydraulic plunger pump under regulated work-ing conditions. In addition, the basic mechanisms of noiseand vibration in a hydraulic plunger pump were revealed.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This research was supported by China Natural Science Foun-dation Project (Grant no. 51135003), the National Key Devel-opment Programme for Fundamental Research (973 Pro-gramme, Grant no. 2014CB046303), and Australian ResearchCouncil (ARC DP150102751).

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