An extraction method for pressure beat vibration characteristics of hydraulic drive system based on variational mode decomposition

QIAN Duo-zhou，GU Li-chen，YANG Sha，MA Zi-wen

(School of Mechatronic Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China)

Abstract：In the pump-controlled motor hydraulic transmission system, when the pressure pulsation frequencies seperately generated by the pump and the motor are close to each other, the hydraulic system will generate a strong pressure beat vibration phenomenon, which will seriously affect the smooth running of the hydraulic system.However, the modulated pressure signal also carries information related to the operating state of the hydraulic system, and a accurate extraction of pressure vibration characteristics is the key to obtain the operating state information of the hydraulic system.In order to extract the pressure beat vibration signal component effectively from the multi-component time-varying aliasing pressure signal and reconstruct the time domain characteristics, an extraction method of the pressure beat vibration characteristics of the hydraulic transmission system based on variational mode decomposition(VMD)is proposed.The experimental results show that the VMD method can accurately extract the pressure beat vibration characteristics from the high-pressure oil pressure signal of the hydraulic system, and the extraction effect is preferable to that of the traditional signal processing methods such as empirical mode decomposition(EMD).

Key words：hydraulic drive system; pressure beat vibration; variational mode decomposition(VMD); characteristic extraction

0 Introduction

The hydraulic drive system of pump-controlled motor is commonly used for metallurgical machinery manufacturing in the field of aerospace engineering due to its high power density and volumetric efficiency.Because the plunger pump and plunger motor have high working pressure and compact structure, they are most commonly used[1].Owing to the structure of plunger pump and plunger motor, flow pulsation will be generated in the working process, resulting in pressure pulsation[2], which is a part of the three properties of plunger equipment.The study shows that when the pressure pulsation frequencies of the pump and motor are close to each other, the system itself has a certain stiffness.Therefore, the hydraulic system will produce pressure beat vibration.The pressure beat makes the pulse amplitudes of the pump and the motor superpose and then the vibration is intensified, which will affect the quality of the hydraulic system.However, the modulated pressure signal also carries information about the system status[3].Therefore, in recent years, researchers have conducted extensive research and in-depth discussion on the pressure vibration phenomenon of hydraulic system.Gao et al.defined the conditions for the vibration of air hydraulic pipeline and proposed a new technology of damping vibration reduction of pipeline system by using viscoelastic constraint layer material[4]；Through simulation and test, Jiang et al.analyzed the law of hydraulic system pressure beat vibration and proved that reasonable selection of transmission ratio and number of plunger could effectively suppress the hydraulic system pressure beat vibration[5]；When the condensate booster pump in the condensate system of Qinshan Nuclear Power was suspected to be vibrating, Li et al.conducted vibration test on the condensate booster pump and its attached pipelines, and determined that the condensate booster pump had slapping vibration by studying the corresponding vibration characteristics of the condensate booster pump[6].The current research on hydraulic system pressure beat vibration mainly focuses on how to suppress it, but pressure beat vibration is rarely used as a carrier to amplify the running state of the hydraulic system.

The pressure beat signal can be regarded as the gain signal of pump and motor pressure pulsation.Because pressure beat signal contains rich information about the running state of the hydraulic system, accurate extraction of pressure vibration signal characteristics is the premise of monitoring the running state of the hydraulic system.With the development of modern industry, high-speed and high-pressure energy saving has become the development trend of hydraulic system, and variable speed pump motor system has become one of the typical representatives of hydraulic drive system[7].This brings many challenges to the characteristic extraction of the pressure beat signal.The variable speed input of the pump is realized by the frequency converter, which brings strong noise interference to the pressure vibration signal；At the same time, due to the variable speed of the pump, the pressure vibration signal of the hydraulic system is a time-varying non-stationary limit number.When the time-varying non-stationary signal is processed by the traditional time-frequency analysis method, there are some shortcomings such as mode aliasing and poor adaptability[8-9].Therefore, the required signal components cannot be decomposed from the complex signal.

As a new time-frequency analysis method[10], variational mode decomposition can decompose multi-component signals into multiple single-component amplitude modulation(AM)and frequency modulation(FM)signals simutaneously, which avoids endpoint effect and false component problems in an iterative process.This method can effectively deal with nonlinear non-stationary signals with extreme noise, therefore it is widely used in fault diagnosis of rotating machinery[11-13].In this paper, we propose a variable mode decompesition(VMD)-based method for extracting the characteristics of hydraulic drive system.It can accurately separate the beat-vibration components from the complex pressure signals and provide technical method support for extracting the operating status information carried by the hydraulic system pressure vibration, monitoring the local status of the hydraulic system and analyzing the global characteristics of the hydraulic system.

1 Principle

1.1 Basic principle of hydraulic system pressure vibration

In a hydraulic drive system, the plunger pump spindle rotates the cylinder and plunger.Under the inclined angle of the inclined plate, the volume of a single plunger cavity is changed according to a sinusoidal law.The spindle rotates for one week, the cylinder body completes oil absorption and discharge once, and the mechanical energy is converted into hydraulic energy.Plunger motors operate in an opposite way.

The volume of the plunger cavity changes periodically as the spindle rotates.In the oil absorption zone, the volume of the plunger cavity increases but the pressure decreases, which creates a vacuum to inhale the oil.In the oil discharge area, the volume of the plunger cavity decreases but the pressure increases, thus the oil is discharged.The actual flow pulsation of the pump is a complicated process with a periodic change due to the influence of oil recirculation and compressibility of the vibration absorber at the distribution port[14].Similarly, the oil flow rate of the hydraulic motor also changes periodically with the load speed.

The Fourier transforms of the instantaneous flows of the pump and motor are completed seperately.Since the flow characteristic of the fundamental wave is very close to that of the original signal, only the fundamental wave is retained to simulate the flow rate pulsation[15].At this time, the flow pulsation can be regarded as a simple harmonic motion, and the flow expressions of pump and motor can be expressed as

Q=Q0+qsin(4ωzt),

(1)

whereQis the instantaneous flow of the pump or motor；Q0is the constant flow of the pump or motor；zis the number of plungers in drainage area; andqis the fluctuation amplitude of the pump or motor flow.

The closed space composed of pump, motor and tubing is studied.The basic principle of fluid mechanics is expressed as

(2)

whereQpis the instantaneous flow of the plunger pump,Qmis the instantaneous flow of the plunger motor,Ctis the leakage coefficient of the hydraulic system,p0is the steady-state working pressure of the pump-controlled motor system,Vis the volume of the closed cavity,Eis the elastic modulus of oil volume, andpis the pressure of closed oil chamber.

Assuming that the system is in an ideal state, that is, the inflow flow of the plunger cavity is equal to the outflow flow, the ideal flow continuity equation between the pump and the motor can be expressed as

Qp0-Qm0-Ctp0=0,

(3)

whereQp0is the steady-state average flow of plunger pump, andQm0is the steady-state average flow of plunger motor.

Substituting Eqs.(1)and(3)into Eq.(2), there is

(4)

whereqmis the pulsation amplitude of the instantaneous flow of plunger motor；ωmis the angular velocity of motor cylinder；zmis the number of plungers in a plunger motor；qpis the pulsation amplitude of the instantaneous flow of plunger pump；ωpis the angular velocity of plunger pump；zpis the number of plungers in a plunger pump.

It can be seen that the flow pulsations of the plunger pump and plunger motor cause pressure pulsation of the pump and motor, which results in pressure pulsation in the hydraulic system.

According to Eq.(4), the pressure pulsation of the hydraulic system can be expressed as

(5)

where

According to Eq.(5), the pressure pulsation of the hydraulic system is actually the superposition and mutual influence of the pressure pulsation signals of the plunger pump and the plunger motor.This pulsation forms new vibration periods and amplitudes.

Due to the leakage of the hydraulic system, when the pump outputs the flow to the motor, the flow is reduced.Therefore, the frequency and amplitude of the motor pressure pulsation are slightly less than that of the pump[16].Assuming that the pressure pulsation frequency generated by the pump source is 220 Hz and the pulsation amplitude is 5.5 MPa, the pressure pulsation frequency generated by the motor is 200 Hz, and the pulsation amplitude is 5 MPa.The resultant pressure pulsation is shown in Fig.1.At this time, the pulsation amplitude of the system is 10.5 MPa and the frequency is 20 Hz.

Fig.1 Numerical simulation waveform and superimposed waveform of pressure pulsation

It can be seen from Fig.1 that the system pressure changes periodically with time, and the hydraulic system experiences obvious pressure vibration.The maximum pressure pulsation of the system is the sum of the pulsation amplitude of the pump and the motor, and the frequency is the difference between the two.Increased pulsation has a significant impact on hydraulic components and shortens their service life.But it also carries more understandable information about system performance.

1.2 Extracting principle of VMD

VMD method is a new variable scale signal processing method proposed by Konstantin Dragomiretskiy in 2014.The method can decompose multi-component AM and FM signals into multiple single-component AM and FM signals at one time[10].It can decompose the complex signalx(t)into a series of essential mode functions, and each essential mode function fluctuates around its central frequency[17].Through non-recursive decomposition, VMD can effectively avoid many problems such as boundary effect.

Supposing that signalx(t)are composed of several components with different center frequencies and limited bandwidths the decomposition problem of signalx(t)can be converted to the variational model for decomposition.Under the constraint that the sum of components is equal to signalx(t), the sum of the bandwidths of all elemental mode functions is minimized.Constrained variational problems can be expressed as

(6)

whereuk={u1,u2,…,uK} is the set of all modal functions,ωk={ω1,ω2,…,ωk} is the frequency set of all centers, ?tis the partial derivative of time with respect to a function,δ(t)is the unit impulse function, j is an imaginary unit, and * stands for convolution operator.

To solve the above equation, by introducing a quadratic penalty function and a Lagrange multiplier, the VMD algorithm can be transformed into an unconstrained optimization problem as

ζ({uk},{ωk},λ)=

(7)

where the quadratic penalty factor can guarantee the reconstruction accuracy of the signal, while the Lagrange multiplier can guarantee the strictness of the constraint condition.

The multiplication operator is used to further solve Eq.(7)in alternating directions, and the eigen-mode functionuk(t)and its corresponding center frequencyωkare obtained by

(8)

(9)

Therefore, the complete VMD algorithm is described as follows.

4)Repeat steps 2 and 3 until iteration constraints satisfy

(10)

whereεis the decomposition condition parameter, usually which isε=1×10-7.

2 Test platform

The schematic diagram of the hydraulic test platform with variable speed of the pump-controlled motor is illustrated in Fig.2.

1—Inverter; 2—Asynchronous motor; 3—Transfer box; 4—Speed and torque sensor; 5—Main pump; 6—oil filling pump; 7—Oil filling overflow valve; 8—Oil filling one-way valve; 9—Electromagnetic proportional overflow valve; 10—Combined sensor; 11—Flushing valve set; 12—Variable motor; 13—Gear pump; 14—Electromagnetic relief valve; 15—Globe valve; 16—Filter; 17—Cooler; 18—Drive system tank; 19—Load system tank

The test platform mainly includes the monitoring system, power source, hydraulic transmission system and loading system.The monitoring system consists of a sensor, a data multifunction card and an industrial computer.The power source is composed of inverter 1 and asynchronous motor 2.By adjusting the control voltage of inverter 1 through the measurement and control system, the variable frequency speed regulation of asynchronous motor 2 can be realized.The plunger pump 4 and the plunger motor 11 constitute a hydraulic transmission system.Both end to end constitute a typical closed system.The plunger pump is a swap-type axial plunger pump, and the plunger motor is a swap-type axial plunger motor.The specific technical parameters are given in Table 1.The loading part of the test platform is an open hydraulic circuit composed of gear pump 12 and electromagnetic proportional relief valve 8.By controlling proportional overflow valve 8, the pressure is generated in the outlet pipeline of gear pump 12.Thus the load torque can be generated on the motor shaft.The physical map of the hydraulic test platform with variable speed of the pump-controlled motor is illustrated in Fig.3.

Table 1 Technical parameters of plunger pump and motor

Fig.3 Physical map of hydraulic test platform with variable speed of pump control motor

In order to accurately obtain the pressure information during the operation of the variable speed hydraulic test platform, pressure sensors of HYDAC hda4844-a-400-y00 were installed at the inlet of the pump, the outlet of the pump, the high-pressure oil line of the system and the inlet of the motor, respectively.Installation position of a pressure sensor for high-pressure tubing is shown in Fig.4.

Fig.4 Installation position of pressure sensor

The main technical parameters of the pressure sensor are shown in Table 2.

Table 2 Main technical parameters of pressure sensor

3 Experiment and analysis

3.1 Acquisition of beat signal

In engineering practice, when the frequency ratio of two vibration sources approaches 20%, the system will beep[18].In the hydraulic system, when the frequency ratio of pump and motor pump oil is similar, that is, 1≤max(i,1/i)≤1.2, the hydraulic system will have the phenomenon of pressure beat vibration.The inequality 0.83≤i≤1.2 can be obtained by calculation.Since a seven-plunger motor and a nine-plunger pump are installed on the test platform, the displacement of the pump is adjusted to 55 mL/r, the displacement of the motor is 70 mL/r, and the ratio of pump oil frequencies of pump to motor is between 0.83 and 1.2.

By adjusting the loading of proportional overflow valve, the system pressure reaches 13 MPa, and then the motor speed slowly increases.When pump speed reaches 1 200 r/min and the motor speed is close to 885 r/min, the pressure waveform of high-pressure oil pipeline is obtained, as shown in Fig.5.It can be seen that the time domain waveform of the pressure signal of high-pressure oil circuit is disordered due to the influence of the interference signal.The signal is transformed by fast Flourier transform(FFT).The corresponding spectrum can be obtained, as shown in Fig.6.

Fig.6 FFT spectrum of pressure signal of high pressure oil line

Fig.5 Time domain waveform of high pressure oil circuit

Signal spectrum components are complicated, which mainly include pump rotation frequency and each harmonic, motor rotation frequency and each harmonic, power frequency interference, noise and so on.After calculation, the rotation frequencyfof the pump is 20 Hz, and the oil pumping frequency isfp=7f=140 Hz.The rotation frequencyfrof the motor is 14.84 Hz, and the pumping frequency isfm=9fr=133.6 Hz.The beat and other aliasing components in the pressure signal cannot be distinguished by FFT.

3.2 Decomposition of VMD

3.2.1 Determination of VMD parameters

The VMD method requires the preset scale numberKand the bandwidth parameters[19].Theoretically, since the center frequency of the band-limited intrinsic mode function(BIMF)component obtained by VMD decomposition is distributed from low frequency to high frequency, ifKjust reaches the optimal value from small to large, the center frequency of the last BIMF component will reach the maximum value for the first time.Therefore, we determine the value ofKby taking the observation center frequency as the maximum value for the first time, andαadoptes the VMD default value of 2 000.WhenKtakes different values, the collected pressure signals are decomposed into VMD, and the center frequency of BIMF component is obtained, as showed in Table 3.

Table 3 BIMF component center frequency of pressure signal

It can be seen from Table 3 that whenK=4, BIMF4 takes the maximum center frequency 390.6 Hz, which is approximately equal to 398.4 Hz(whenK=5, BIMF5 takes the maximum center frequency).At the same time, it is obvious whenK=5, there is over-decomposition, that is, modal components of the same center frequency appear(BIMF3 and BIMF4), thereforeK=4.

3.2.2 Extraction of time-domain characteristics of pressure beat vibration

VMD method was adopted to decompose the collected high-pressure oil circuit pressure signals.The numberKof mode decomposition was set at 4, and the penalty factor was set at 2 000.The breakdown diagram is shown in Fig.7.

Fig.7 VMD decomposition diagram of pressure signal in high-pressure oil circuit

It can be clearly seen that the time-domain waveform of componentu1has an obvious amplitude modulation phenomenon.The waveform is very consistent with the beat vibration curve obtained by theoretical analysis.u1is amplified, and the waveform obtained is shown in Fig.8.After FFT transformation, the spectrum diagram obtained is shown in Fig.9.

Fig.8 Time-domain enlargement of component u1

Fig.9 FFT spectrum of component u1

It can be observed from Fig.8, that the period of beatings is about 0.17 s and the frequency of beatings is 5.9 Hz.The theoretical calculation shows that the difference of one pump oil frequency between the pump and motor is 6.4 Hz, which is basically consistent with the derivation conclusion that the beat wave frequency is the difference between the original two vibration frequencies in the beat theory.In Fig.9, there are obvious peak values at 133.7 Hz and 139.4 Hz.The frequency corresponds to one-time frequency of the motor and the pump respectively.The interval frequency between the motor and the pump is 5.7 Hz, which is consistent with the theoretical conclusion of beat vibration.

It can be seen from Fig.10 that EMD decomposition fails to effectively extract the beat vibration component.

Fig.10 Pressure signal EMD decomposition diagram of high-pressure oil circuit

In Fig.11, IMF1 contains frequency components of 139.4 Hz and 280 Hz, IMF2 is mixed with frequency components of 280-699.4 Hz, and IMF3 is composed of frequency components of 532 Hz and 699.4 Hz.The frequency amplitude of pump and motor oil extracted by EMD method is far less than that of VMD method.The results show that the decomposition effect of EMD algorithm is not ideal and the modal aliasing is serious for the complex pressure signal with time-varying high noise.

Fig.11 Spectrum diagram obtained after EMD decomposition

The comparison verifies the validity of VMD in time domain feature extraction of pressure beat vibration of the hydraulic system.

4 Conclusions

1)In a pump-controlled motor hydraulic drive system, when the pulsation frequencies of the pump and motor are close to each other, the system pressure beats.Vibration makes the noise of hydraulic system increase, and the service life of components decreases.

2)The proposed method can effectively separate the vibration components of mixed pressure signals in complex pipeline.Experimental results show that this method is better than the traditional signal processing method in extracting the characteristics of time-varying high-noise signal.

3)The time-domain characteristics of the hydraulic system pressure vibration can be extracted effectively by VMD method.The method provides a method support for quantitative analysis of the operating state information carried by the hydraulic system pressure vibration and extraction of time-varying non-stationary signals of the hydraulic system.At the same time, it has certain application value for monitoring the operating state of the hydraulic system.