Research Article Vibration Modes And The Dynamic Behaviour Of . - Hindawi
Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 9679542, 7 h ArticleVibration Modes and the Dynamic Behaviour ofa Hydraulic Plunger PumpTianxiao Zhang and Nong ZhangSchool of Electrical, Mechanical and Mechatronic Systems, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, AustraliaCorrespondence should be addressed to Tianxiao Zhang; xiaozhang717@126.comReceived 19 August 2015; Revised 19 November 2015; Accepted 22 November 2015Academic Editor: Vadim V. SilberschmidtCopyright 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 vibration mechanisms in hydraulic pumps. The workingconditions 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-offreedom system were 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. IntroductionSince there have been increasingly higher requirementsimposed on the engineering quality, the accuracy, and reliability 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 [1β3].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 reducing 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. Vibration 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-structure interaction would be caused between solid and liquidunder different conditions. The field of hydraulic model andvibration has been discussed in some research [4β7]. 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 [8β16], 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 environment 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
2Shock and VibrationVibrationtheory HydraulictechnologyHigh speed(i) Avoid thebreakdown or failureof hydraulic systemand deviceHigh powerHigh pressurePrecise treatmentVibration analysis on the hydrauliccomponent and systemLight weightIntellectualizationHigh reliabilityHumanizationVibration theory of solidVibration theory of fluidVibration theory of FSIEnvironment-friendly(ii) Reduce thevibration noisepollution of hydraulicsystem and device(iii) Improve theperformance ofhydraulic system anddeviceFigure 1: Vibration analysis: hydraulic component and system.k3F2 (t)x2 (t)k2m2c3F1 (t)x1 (t)k1m1c2c1Figure 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, the model reveals the basic mechanisms of vibration 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 Pump2.1. 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 π1 (kg), and themasses of its retainer plate, seven plungers, and sliding shoeswere merged and represented by π2 (kg). The connectionstiffness and the damping between the cylinder block andport plate were, respectively, set to π1 (N/m) and π1 (N s/m).The cylinder and retainer plate were connected by a centralspring whose stiffness and damping were π2 (N/m) and π2(N s/m), respectively. The sliding shoes were connected withan angle plate with a connection stiffness and damping of π3(N/m) and π3 (N s/m), respectively.According to the proposed vibration model, the positivedirections of the acceleration and excitation were determined,which were in accordance with the positive direction of thecoordinate axes. The deduced vibration differential equationisπ1 π2 π2π₯Μ 1π₯Μ 1π1 0]{ } []{ }[0 π2 π₯Μ 2 π2 π2 π3 π₯Μ 2(1)π₯1πΉ1 (π‘)π1 π2 π2]{ } { [}, π2 π2 π3 π₯2 πΉ2 (π‘)where the constant matrices including M [π], C [π], andK [π], which are constituted by coefficient matrices, are,respectively, the mass matrix, damping matrix, and stiffnessmatrix: the vector x {π₯} 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 analysed to study pressure pulsation. Flow pulsation refers tothe instantaneous flow variation when a hydraulic pump isrunning. When a hydraulic pump keeps operating continuously, the ever-changing sealing volume is expected to begenerated in a majority of hydraulic pumps. Meanwhile, theinstantaneous flow changes repeatedly, and some instantaneous, nonconstant flow may generate flow pulsation. Theinstantaneous actual flow of a hydraulic plunger pump maybe given as follows.When 0 π πΌ and π§0 (π§ 1)/2,cos (πΌ/2 π)2 sin (πΌ/2)2ππ π πππ tan πΎ cos (πΌ/2 π) .8 sin (πΌ/2)ππ 1 π΄π πππ tan πΎ(2a)When πΌ π 2πΌ and π§0 (π§ 1)/2,cos (3πΌ/2 π)2 sin (πΌ/2)2ππ π πππ tan πΎ cos (3πΌ/2 π) ,8 sin (πΌ/2)ππ 2 π΄π πππ tan πΎ(2b)
Shock and Vibration3where π΄ represents the cross-sectional area (m2 ) of theplunger; π is the radius (m) of the central circle of the plunger,and π indicates the diameter (m) of the plunger; πΎ denotesthe inclination angle (rad) of the angle plate, π§ is the numberof the plungers, and π is the angular velocity (rad/s) of theoil cylinder block; π is the rotational speed (rpm) of theplunger pump, ππ is the volumetric efficiency, and πΌ is thehalf included angle (rad) between two adjacent plungers andπΌ π/π§; π represents the angular displacement (rad) of thecylinder block, and π ππ‘. LetπΎπ ππ2 π πππ tan πΎ ππ2 π πππ tan πΎ .8 sin (πΌ/2)8 sin (π/2π§)(3)pump inevitably leads to a pressure change. With respect toa compressible flow, its instantaneous pressure can be represented by the following formulas.When 0 π‘ π/π§π,ππ 1 πΈπ7ππ3πΌππ 2 πΎπ cos ( π) .2(4b)π)2π§in case 0 π‘ ππ 2 πΎπ cos (ππ‘ 3π)2π§in caseπ,π§πππ0π) dπ‘2π§(8a)When π/π§π π‘ 2π/π§π, Converting the instantaneous actual flow into a functionof time π‘(π ), thenππ 1 πΎπ cos (ππ‘ 0π‘ cos (ππ‘ ππ[sin (ππ‘ ) sin ( )] . 7ππ π2π§2π§ππ 2 (4a)πΈπ πΎππΈπ πΎπThenπΌππ 1 πΎπ cos ( π) ,2π‘ ππ 1 dπ‘ πΈπ7ππ πΈπ πΎπ7πππΈπ πΎπ7ππ ππ‘π/π§π[ ππ 2 dπ‘π‘π/π§πcos (ππ‘ [sin (ππ‘ 3π) dπ‘]2π§(8b)3ππ) sin ( )] ,2π§2π§(5b)where πΈπ represents the elastic modulus of the flow (Pa) andthe volume of the closed cavity for a single plunger is ππ π΄ π π (ππ2 /2)π tan πΎ.where ππ 1 and ππ 2 represent the instantaneous flow throughthe plunger pump. It can be seen that when π πΌ/2 and π 3πΌ/2 (namely, π π/2π§ and π 3π/2π§), instantaneous flowwas at a maximum, ππ max ; when π 0 and π πΌ (namely, π 0 and π π/π§), instantaneous flow was at a minimum, ππ min .Therefore, the flow pulsation coefficient πΏπ can be defined asfollows:2.4. Fourier Series Simulation. The flow and pressure pulsation in a hydraulic plunger pump are a nonharmonic periodicfunction, which can be represented by a harmonic Fourierseries. Thereby, the dynamic response problem corresponding to the harmonic excitation of its Fourier series can besolved. The periodic excitation function in an interval of[π1 , π2 ] is unfolded as the Fourier series; namely,πΏπ π2π π‘ ,π§ππ§πππ max ππ min,ππ‘(5a)(6)πΉ (π₯) where ππ‘ is the theoretical mean flow. When the cylinderblock of the plunger pump rotates through one completerevolution, each plunger moves back and forth once in a cycleof oil adsorption and extrusion. Therefore, the theoreticalmean flow ππ‘ and the actual flow π can be presented asππ‘ 2π΄π§ππ tan πΎ 15π§ππ2 π tan πΎ,π 2π΄π§ππ ππ tan πΎ 15π§ππ2 π ππ tan πΎ.ππ (2π₯ π1 π2 )π0 (π cos2 π 1 ππ2 π1 ππ sinππ (2π₯ π1 π2 )π) 0π2 π12(9) [ππ cos π (π§ππ₯ π) ππ sin π (π§ππ₯ π)] ,π 1(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 plungerwhere π1 0 and π2 2π/π§π. Equation (9) shows thata complex periodic excitation function can be decomposedinto the superposition of a series of harmonic functions. Thecoefficients of Fourier series π0 , ππ , and ππ can be determinedas follows:ππ π2ππ (2π‘ π1 π2 )2dπ‘ πΉ (π‘) cosπ2 π1 π1π2 π1π§π 2π/π§ππΉ (π‘) cos π (π§ππ‘ π) dπ‘ π 0(π 0, 1, 2, 3, . . .) ,(10a)
4Shock and Vibrationππ π2ππ (2π‘ π1 π2 )2dπ‘ πΉ (π‘) sinπ2 π1 π1π2 π1π§π 2π/π§ππΉ (π‘) sin π (π§ππ‘ π) dπ‘ π 0(10b)(π 1, 2, 3, . . .) .The function πΉ(π₯) can be represented by Fourier seriesas long as the defined integrations of ππ and ππ actually exist.If πΉ(π₯) 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 is2πΈπ πΎπππ0 sin ( ) ,(11a)27ππ π2π§ππ π/π§ππ§π πΈπ πΎπππ[sin (ππ‘ ) sin ( )][ π 7ππ π 02π§2π§ cos π (π§ππ‘ π) dπ‘] 2π/π§ππ/π§π[sin (ππ‘ (11b) 2π/π§ππ/π§π[sin (ππ‘ sin π (π§ππ‘ π) dπ‘ sin (π§π πΈπ πΎππ 7ππ π3. Vibration of the Two-Degree-of-FreedomSystem of a Hydraulic Plunger Pump(11c)1π§π πΈπ πΎπ{ π 7ππ π(1 ππ§) πππ1sin ( ππ)} ππ) 2π§2π§(1 ππ§) π(π 1, 2, 3, . . .) .Therefore, the quantity of pressure pulsation induced bythe quantity of flow pulsation can be expressed as πππ (π‘) 0 ππ sin π (π§ππ‘ π)2 π 1(13b)where π΄ π is the cross-sectional area of a single plunger andhere π΄ πΆ 7π΄ π ; therefore, the excitation πΉ2 (π‘) is consideredto be equal to πΉ1 (π‘), where πΉ1 (π‘) and πΉ2 (π‘) are a pair of actionand counteraction forces with the same magnitude but actingin opposite directions.(π 1, 2, 3, . . .) ,3ππ) sin ( )]2π§2π§(13a)where π΄ πΆ represents the cross-sectional area of the cylinderhole of the cylinder block, π΄ πΆ 7(π/4)π2 , and π is thediameter of piston. ConsiderπΉ2 (π‘) 7ππ π΄ π ,π§π πΈπ πΎπ π/π§πππ[sin (ππ‘ ) sin ( )] π 7ππ π 02π§2π§ sin π (π§ππ‘ π) dπ‘ 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 byπΉ1 (π‘) ππ π΄ πΆ,3ππ) sin ( )]2π§2π§ cos π (π§ππ‘ π) dπ‘ 0ππ π§π πΈπ πΎππ 7ππ πFrom 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 π₯-axis, negative value of vertical axis expresses thedirection of pulsation quantity which is opposite to π₯-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 harmonic, 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.(π 1, 2, 3, . . .) , (12)where ππ is the quantity of pressure pulsation of the plungerpump. The sum of pressure pulsation and mean pressureexpresses the pressure work in pump; that is, π(π‘) π0 ππ (π‘).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 π (π 1, 2, . . . , 10)with its different values.3.1. The Natural Frequency and Modes of Vibration of theSystem. As seen from (1), matrices M [π], C [π],and K [π] are symmetrical, and the matrix elements are,respectively, listed as follows:π11 π1 ,π12 π21 0,π22 π2 ,π11 π1 π2 ,π12 π21 π2 ,π22 π2 π3 ,π11 π1 π2 ,π12 π21 π2 ,π22 π2 π3 .(14)
Shock and Vibration5Therefore, the constants in pairs such as π’1(1) and π’2(1) andand π’2(2) can determine the natural vibration modes ormain vibration mode displayed by the system when the system conducts synchronous harmonic motion, respectively, atfrequencies of π1 and π2 . These constants can be expressedby the following matrices: 1075π’1(2)432ps (Pa)1u(1) {0 1 2u(2) { 3 4 500.0050.010.0150.020.0250.03t (s)j 1j 2j 3j 5j 10Figure 3: Pressure pulsation ππ (Pa).To study the inherent characteristics of a hydraulicplunger pump, the deduced characteristic equation, or frequency equation, of the two-degree-of-freedom system of ahydraulic plunger pump may be given as follows:Ξ (π2 ) π1 π2 π4 (π1 π22 π2 π11 ) π2 π11 π222 π12 0.(15)Equation (15) is a quadratic equation in π2 , and its rootsareπ12π22 1 π1 π22 π2 π112π1 π22π π π12π π π2 π11 21 ( 1 22) 4 11 22.2π1 π2π1 π2(16)2Since π11 π22 (π1 π2 )(π2 π3 ) π22 π12, asseen in (16), the value of the item following the sign wassmaller than that to the left of it, and thus π12 and π22 arepositive numbers. Therefore, two positive real roots of thecharacteristic equation in (15) can be obtained, namely, thetwo natural frequencies π1 and π2 of the system.π’1(1) and π’2(1) are used to represent the amplitudes corresponding to π1 , and π’1(2) and π’2(2) are the amplitudescorresponding to π2 . The homogeneous differential equationcan only determine the ratios of π’2(1) /π’1(1) and π’2(2) /π’1(2) . Thededuced amplitude ratios are shown as follows:π1 π2 π’2(1)π’1(1)π’2(2)π’1(2) π11 π12 π1π12 ,π12π22 π12 π2(17a) π11 π22 π1π12 .π12π22 π22 π2(17b)π’1(1)1(1)} π’{},1π1π’2(1)π’1(2)1(2)} π’{}.1π2π’2(2)(18a)(18b)There are two natural frequencies in this system, andcorrespondingly there are two natural vibration modes.The lower frequency π1 is the first-order natural frequencyor, simply, the fundamental frequency, while the higherfrequency π2 is the second-order natural frequency. Thecorresponding vibration modes 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 improvement and development of computer software, hardware, 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 differential 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-π,and (4) Newmark-π½.4. A Numerical ExampleThe mass of the cylinder block of a certain hydraulic plungerpump was π1 2.7 kg, and the masses of the retainer plate,seven plungers, and its sliding shoes were merged togetheras π2 1.8825 kg. Suppose that the connection stiffnessand damping between the cylinder block and the port platewere, respectively, π1 3.6 107 N/m and π1 329.0 N s/m,and the stiffness and damping between the cylinder and theretainer plate were π2 5.0 105 N/m and π2 223.0 N s/m,respectively; the stiffness and damping between the slidingshoes and the swash plate were, respectively, π3 7.7 107 N/m and π3 490.0 N s/m. The inclination angle of theswash plate was πΎ 19.5 , the radius of the central circle ofthe plunger was π 0.285 m, the diameter was π 0.019 m,
6Shock and Vibrationand number of the plungers was π§ 7. Meanwhile, the rateof rotation of the plunger pump was π 1500 rpm. Theelastic modulus of the flow and the volumetric efficiency ofthe plunger pump were, respectively, πΈπ 1.226 103 MPaand ππ 0.95. This research attempted to obtain the naturalfrequency and vibration modes of the system as well as itsvibration response.According to the known conditions, here we haveu(1)π110.0096u(2)π21π11 π1 π2 ,π12 π21 π2 , 149.3203(19)π22 π2 π3 .By substituting the above into (16), the natural frequencycould be obtained:1π1 3676 ,s1π2 6416 .s(20)Figure 4: Natural mode of vibration of the two-degree-of-freedomsystem. 10 36The values of π1 and π2 can be obtained by substituting π12 andπ22 into (17a) and (17b):π2 149.3203.42(21)The natural modes of vibration for the system can becalculated according to (18a) and (18b):x (m)π1 0.0096,x10 21},u(1) {0.00961}.u(2) { 149.3203x2 4(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-π½ method was applied to the vibration modelhere, and the values of π½ and πΏ were taken, respectively, asπ½ 1/4 and πΏ 1/2. The displacement π₯1 (m) of mass π1and displacement π₯2 (m) of mass π2 over time were as shownin Figure 5.As discovered during calculation, owing to the unstableinitial 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, compared with the retainer plate, seven plungers, 600.0050.010.015t (s)0.020.0250.03Figure 5: The response of the two-degree-of-freedom system withtime π‘.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. ConclusionsThe mechanical vibration analysis and the dynamic design arekey points during the engineering design of mechanical products 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 characteristics 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
Shock and Vibrationmodel for the dynamic analysis of a two-degree-of-freedomhydraulic plunger pump system. Meanwhile, the inherent characteristics and dynamic response of the hydraulicplunger pump were studied and were used to estimate thevibration of a hydraulic plunger pump under regulated working conditions. In addition, the basic mechanisms of noiseand vibration in a hydraulic plunger pump were revealed.Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper.AcknowledgmentsThis research was supported by China Natural Science Foundation Project (Grant no. 51135003), the National Key Development Programme for Fundamental Research (973 Programme, Grant no. 2014CB046303), and Australian ResearchCouncil (ARC DP150102751).References[1] S. Timoshenko, D. H. Young, and W. Weaver Jr., Vibration Problems in Engineering, John Wiley & Sons, New York, NY, USA,4th edition, 1974.[2] L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill,New York, NY, USA, 1975.[3] F. S. Tse, I. E. Morse, and R. T. Hinkle, Mechanical VibrationsTheory and Applications, Allyn and Bacon, Boston, Mass, USA,1978.[4] W. Sochacki, βModelling and analysis of damped vibration inhydraulic cylinder,β Mathematical and Computer Modelling ofDynamical Systems, vol. 21, no. 1, pp. 23β37, 2015.[5] D. Cekus and B. Posiadala, βVibration model and analysis ofthree-member telescopic boom with hydraulic cylinder for itsradius change,β International Journal of Bifurcation and Chaos,vol. 21, no. 10, pp. 2883β2892, 2011.[6] A. Turnip, K.-S. Hong, and S. Park, βModeling of a hydraulicengine mount for active pneumatic engine vibration controlusing the extended Kalman filter,β Journal of Mechanical Scienceand Technology, vol. 23, no. 1, pp. 229β236, 2009.[7] B. P. Bettig and R. P. S. Han, βModeling the lateral vibrationof hydraulic turbine-generator rotors,β Journal of Vibration andAcousticsβ Transactions of the ASME, vol. 121, no. 3, pp. 322β327, 1999.[8] I. K. Dulay, Fundamentals of Hydraulic Power Transmission,Elsevier Science Publishers, Amsterdam, The Netherlands,1988.[9] M. Noah, Hydraulic Control Systems, John Wiley & Sons,Hoboken, NJ, USA, 2005.[10] A. Akers, M. Gassman, and R. Smith, Hydraulic Power SystemAnalysis, CRC Press, Boca Raton, Fla, USA, 2006.[11] J. Houghtalen Robert, A. A. Osman, and H. C. Hwang Ned,Fundamentals of Hydraulic Engineering Systems, Prentice Hall,London, UK, 4th edition, 2009.[12] P. Kumar, Hydraulic Machines: Fundamentals of HydraulicPower Systems, CRC Press, Boca Raton, Fla, USA, 2012.[13] R. T. Burton, P. R. Ukrainetz, P. N. Nikiforuk, and G. J.Schoenau, βNeural networks and hydraulic controlβfrom simple to complex applications,β Proceedings of the Institution of7Mechanical Engineers, Part I: Journal of Systems and ControlEngineering, vol. 213, no. 5, pp. 349β358, 1999.[14] R. Spence and J. Amaral-Teixeira, βInvestigation into pressurepulsations in a centrifugal pump using numerical methodssupported by industrial tests,β Computers & Fluids, vol. 37, no.6, pp. 690β704, 2008.[15] B. Zhang, B. Xu, C. L. Xia, and H. Y. Yang, βModeling and simulation on axial piston pump based on virtual prototype technology,β Chinese Journal of Mechanical Engineering, vol. 22, no. 1,pp. 84β90, 2009.[16] B. Xu, J. Zhang, and H. Yang, βSimulation research on distribution method of axial piston pump utilizing pressure equalization mechanism,β Proceedings of the Institution of MechanicalEngineers Part C: Journal of Mechanical Engineering Science, vol.227, no. 3, pp. 459β469, 2013.
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To prevent hydraulic systems from breaking down, analysis of their vibration is required. An e ective method of reduc-ing vibration and noise needs to be developed: its aim is to improve the performance of hydraulic devices and thus reduce the vibration and noise in hydraulic systems. Vibra-tion and noise arise from the interaction between solids
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The ISO 2631 suggests vibration measure-ments in the three translational axes on the seat pan, but only the axis with the greatest vibration is used to estimate vibration severity.5 The current trend in vibration research is to use multi-axis val-ues.
Rotating and reciprocating Mechanical vibration - Evaluation of machine ISO 10816 : 1-2-3-4-5-6-7 Rotating machinery Mechanical vibration - Evaluation of machine vibration by measurements on rotating shafts ISO 7919 : 1-2-3-4-5 Turbine a gas. Misurazioni e valutazione delle UNI 10583 Modal Analysis Vibration and shock - Experimental determination
The book normally used for the class at UIUC is Bartle and Sherbert, Introduction to Real Analysis third edition [BS]. The structure of the beginning of the book somewhat follows the standard syllabus of UIUC Math 444 and therefore has some similarities with [BS ]. A major difference is that we deο¬ne the Riemann integral using Darboux sums and not tagged partitions. The Darboux approach is .
