泵毕业设计外文文献

Mansour A.Karkoub a ,Osama E.Gad a ,Mahmoud G.Rabie b

a College of Engineering and Petroleum,Kuwait University,Kuwait

b Military Technical College,Cairo,Egypt

Received 15October 1996;received in revised form 25March 1998;accepted 23April 1998

Abstract

A neural network model for an axial piston pump (bent-axis design)is derived in this paper.The model uses data obtained from an experimental setup.The purpose of this ongoing study is the reduction of the power loss at high pressures.However,at the beginning,a study is being done to predict the behavior of the current design of the pump.The neural network model has a feedforward architecture and uses the Levenberg±Marquardt optimization technique in the training process.The model was able to predict the behavior of the pump accurately.#1998Elsevier Science Ltd.All rights reserved.

1.Introduction

Variable-displacement axial piston pumps are important devices that are commonly used in ¯uid power systems,such as controlled hydraulic power supplies and controlled hydrostatic transmission drives.The displacement varying mechanism and the power-to-weight ratio of this device make it most suitable for control of high power levels.The design of axial piston pumps boasts dependability and simplicity;however,their most important characteristic is that of variable output displacement.A lot of work has been done in the area of axial piston pumps;however,only a few contributions will be discussed here.Kaliafetis and Costopoulos [5]studied the static and dynamic characteristics of an axial piston variable displacement pump with pressure regulator.The accuracy of the proposed model is dependent on the dynamic operating curves of the manufacturer's data.The authors concluded that the operating conditions are very crucial for the pump dynamic behavior which can be improved by decreasing the set pressure.Harris et al.[4]simulated and measured the cylinder pressure and suction ¯ow ripple for an axial piston Mechanism and Machine Theory 34(1999)1211±12260094-114X/99/$-see front matter #1998Elsevier Science Ltd.All rights reserved.PII:S 0094-114X(98)00086-

X

www.elsevier.com/locate/mechmt

pump.Kiyoshi and Masakasu [7]studied the experimental and theoretical static and dynamic characteristics of the operating moment of a swash-plate type,variable delivery,axial piston pump.A new prediction method of the operating moment was proposed.The validity of the new method was veri®ed experimentally for the pump characteristics in which a port plate with wide,short and deep notches was used.Edge and Darling [2],studied the cylinder pressure and ¯ow in an oil hydraulic axial piston pump.The proposed model was checked experimentally.The in¯uence of the port plate and relief groove design on the cylinder pressure and pump ¯ow ripple was discussed.An alternative modeling technique,neural networks (NN),has been proven to give good results,especially when the system is highly nonlinear.This technique tries to emulate the work of the human brain in recognizing information.Karkoub and Elkamel [6]used a neural network model to predict the pressure distribution in a rectangular gas bearing.The designed model predicted the pressure distribution as well as the load-carrying capacity more accurately than the other available tools.Gharbi et al.[3]predicted the breakthrough oil recovery using neural networks.The NN model performed much better than the common regression or ®nite di erence methods.Li et al.[8]used NNs along with the Powell's optimization technique to model and control single and double link inverted pendulums.The authors were able to achieve good results.Panda et al.[9]applied NNs to model ¯uid contacts in Prudhoe Bay oil ®elds.The resultant model estimated the ¯uid distribution at target oil wells more accurately than the conventional regression-based techniques.Aoyama et al.[1]have derived an NN model to predict the response of non-minimum phase systems.The developed model was applied to the Van de Vuss reactor and a continuous stirred tank bioreactor and the results were promising.This paper deals with the modeling of certain pressures in an axial piston pump (bent-axis design)using neural networks.A description of the experimental setup used to collect the training data will be given ®rst,then,a brief introduction to the neural network modeling procedure will follow.

2.Experimental setup

The training data was obtained from an experimental setup which will be discussed in this section.The main component of the setup is the axial piston pump.In the following section,we will give a description of how the pump works,then a description of how the training data was collected will follow.

2.1.Bent-axis pump

The schematic diagram shown in Fig.2describes the basic components of the axial piston pump used in the experiment while the controller for this pump is shown in Fig.3.The pump consists of two main groups.The ®rst group is the rotary group,which contains the drive shaft (31),pistons (32),cylinder blocks (33),and control lens (34).Seven pistons are ®tted in a spherical arrangement on the front face and these simultaneously set the cylinder in rotation.The cylinder is pressed against the control area of the control lens by means of a spring (35).

M.A.Karkoub et al./Mechanism and Machine Theory 34(1999)1211±1226

1212

When the operating pressure P exceeds the preset value at the spring(41),the control elements(38,39and40)push against it.At the same time,the oil¯ows through ori®ce(45) and(46)from the pump exit cavity.The high pressure oil in the cavity volume V1is thus allowed to¯ow through the opening area(47)to the control piston(37)large area.Should the pressure forces acting on the control piston be greater than the spring force,the control piston (37)is moved until the balance of the hydraulic and mechanical forces is restored.The cylinder,pistons and control lens move on the spherical sliding surface(36)in the opposite direction to reduce the swivel angle a max2.2.Data measurement setup

The model designed using neural networks has to be trained using some real data

obtained Array Fig.1.Photograph of the experimental setup.

from the above described system.The training procedure is necessary for the neural network to learn the model it is trying to predict.The data is collected from the experimental setup shown in Fig.1.This data set is obtained by measuring the steady-state and transient responses of the axial piston pump shown in Fig.2.The experimental investigation is carried out on a testbed,shown in Fig.1and the hydraulic circuit diagram shown in Fig.4.The suction and discharge lines of the test pump are connected directly to the suction and high ¯ow meter ports (24)and (25),respectively.The test pump (16)is driven by means of a high power hydraulic motor of controllable speed (13).The hydraulic circuit operates as follows:the oil ¯ows from reservoir (1)to the inlet port of the booster pump (4).The pressure relief valve (7)is used to protect the booster pump circuit against over pressure.The discharge of the booster pump passes through a non-return valve (6)to the suction and supply lines of the main pump (3).The pressure relief valve (8)is used to protect the main pump circuit against over pressure.The discharge of the main pump passes through the direction control valves (9)and (10)used to control the direction of the main pump discharge to the main drive motor circuit (12).The ¯ow rate of the test pump is indicated on a digital ¯ow meter (20).The speed of the test pump driven shaft is measured via a tachometer (15)and can be controlled by changing the speed of the electrical motor (5).The temperature of the working oil is maintained at 50258C during the operation.During the steady-state measurements,the variation of the supply pressure P is adjusted by means of the control valve (29)while the shut-o valve (30)is fully closed.The pressure gage (23)measures the oil pressure in the suction line and a digital pressure gage (21)indicates the oil pressure in the discharge line.The pressure relief valve (28)is used to protect the test

pump

Fig.2.Schematic diagram of the bent-axis piston pump.

M.A.Karkoub et al./Mechanism and Machine Theory 34(1999)1211±1226

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Fig.3.Schematic diagram of the control unit of the piston pump.

Table 1Pump parameters

Parameter

Description Value A c Large side area of control piston 0.000531m 2A p Piston area 0.000531m 2A pp Area of control element (38)0.0000246m 2A s Small side area of control piston 0.0000785m 2V Volume of pump delivery line 2.6Â10À3m 3V 1Volume of the ®rst control cavity 8.2Â10À6m 3V 2Volume of the second control cavity 1.7Â10À7m 3V 3Volume of the third control cavity 1.6Â10À5m 3a min Minimum cylinder inclination angle 48a max Maximum cylinder inclination angle 238

M.A.Karkoub et al./Mechanism and Machine Theory 34(1999)1211±12261215

circuit against overloading.During the transient measurements,valve (30)is fully open and valve (29)is fully closed.

2.3.Measurement of the steady-state response of the pump

The experimental determination of the steady-state performance of the studied pump is carried out by measuring the pump discharge ¯ow Q p at di erent values of the supply pressure P .The test pump parameters are presented in Table 1.The supply pressure P is controlled by the throttle valve (29)and measured by the digital pressure gage (21).The corresponding pump discharge Q p is measured by the digital ¯ow meter (20).Measurements were carried out for di erent pump speeds,550,800and 1000rpm at the same preset pressure.The pump discharge ¯ow Q p was also measured at di erent values of the preset pressure.The measured values are shown in Figs.7and

8.Fig.4.Schematic diagram of the hydraulic system.

M.A.Karkoub et al./Mechanism and Machine Theory 34(1999)1211±1226

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2.4.Measurement of the transient response of the pump

The experimental determination of the transient response of the studied pump,shown in Fig. 2,is carried out by measuring the operating pressures in di erent control cavities.Three electrical pressure transducers are mounted in di erent positions of the pump house which are connected directly to the control cavities of volume V1,V2and V3,as shown in Fig.3.Another pressure transducer is mounted at the pump exit line of volume V to measure the supply pressure P.The transducers are of piezoresistive type and can measure pressures between0.1 and400bar.The input voltage to each transducer is in the range of10±30V;however,the output voltage is between0and5V.These transducers are used to measure the control pressures P1,P2,P3and P(see Fig.3).A time record of the pressure signals is sampled using a PC and a data acquisition board.The board has12bit successive approximation converter with a12m s conversion time giving a maximum throughput rate of70kHz.A®xed loading ori®ce(18)and direction control valve,DCV,(17)are mounted on the pump exit line(see Figs.1and4).These are used to introduce rapid changes in the pump exit line pressure P. When the solenoid of the DCV is energized,the valve closes rapidly and the pump discharge Q p is forced to¯ow through the loading ori®ce.The testbed is arranged such that the current to the solenoid triggers the data acquisition system,thus picking up the transient variation of the pressures P1,P2,P3and P.These measurements are carried out at di erent pump speeds: 550,800and1000rpm.The measured values are shown in Figs.9±11.

3.Neural networks

In this paper,a computational tool,known as neural networks,is used to predict the behavior of bent-axis piston pump.These networks are nothing but a number of interconnected elements known as neurons.These neurons or processing elements are well

Fig.5.Schematic representation of a single neuron.selected linear or nonlinear functions that process any applied input to a known output.The input to the neurons is a weighted sum of the external inputs or the outputs of the neurons immediately preceding it.A small weight applied to the output of the neuron means that the following neuron does not process the input.In this manner,a speci®c circuit will be established for each pattern or input.This type of reasoning or circuitry makes neural networks capable of capturing nonlinearities often undetected by common modeling techniques.

The output of a speci®c neuron is a function of three main factors:the weighted input,the bias of the neuron,and the transfer function(see Fig.5).The output of any neuron is given by:

a f x b

where

x

w i I i

The transfer function f can be selected from a set of readily available functions.The function chosen for our task is the sigmoidal function:

f x

1 1 eÀx

This function is known to give good results,especially if the outputs are known for the given inputs.

Any network is usually divided into subnetworks or commonly referred to as layers.Each

Fig.6.Schematic representation of a multi-layer,feedforward,neural network.network contains two essential layers which are the input and output layers and one or more hidden layers if the task requires that.Fig.6shows a typical feedforward architecture of a neural network.The output of the output layer is a result of a combined e ect of all the neurons in the network.

3.1.Training

Designing a neural network requires at least four major steps:(1)selecting the number of layers;(2)selecting the number of neurons;(3)selecting the types of transfer functions;and(4) selecting a training data set that captures the behavior of the system.The training process is time consuming and very critical for the success of the network.Several techniques are used to do the training;among these is backpropagation with momentum.The weight of every input to every neuron is updated consecutively starting from the output layer and working backwards.During this process an objective function,usually the sum squared error,is minimized.Several optimization techniques are used in the literature including the Powell's and the Levenberg±Marquardt algorithms.The latter technique is used in this paper.This technique switches between the famous gradient descent(see the Appendix)and the Gauss±Newton algorithms.The so-called update rule in the Levenberg±Marquardt method is given by:

D W À

c T c a I

ÁÀ1

c T E 1

where c is matrix of derivatives of errors to each weight,a is a scalar,and E is the error vector.

3.2.Data selection

One of the factors that make the training process very time consuming is the size and quality of the training data.If the data does not cover all the details about the behavior of the system, the optimization procedure might not converge to the expected answer.However,obtaining a data set that describes all aspects of the system is not an easy task.Moreover,the training data can be redundant,i.e.several patterns convey the same information.Therefore,the training time will be increased to process the same information over and over again.If that is the case, it is suggested that the Karhunen±Loeve decomposition be used.

Suppose that we have a data set f(x,t)with M input vectors.De®ne the average of these vectors X:

"X 1

M M

i 1

X i

let

X

i X iÀ

"X

The covariance matrix is de®ned as follows:C ij 1

M

X

iÁ

X

j

The eigenvalues and eigenvectors of the covariance matrix can be obtained by singular values decomposition.Let f k i be the i th component of the k th eigenvector.The projection of the vectors onto the eigenspace is given by:

c k

i 1

M f k i X i

De®ne the energy of the set E as follows:

E

M

i 1

l i

where l i is the eigenvalue corresponding to the i th eigenvector.The energy of each input vector can be de®ned as follows:

E k l k E

Now,calculate the coordinates a i as follows:

a i

XÁc i c iÁc i

If the energy of the input vector is very small then the point will not have additional information about the system.Suppose that there are N energetic input vectors arranged from the most to least energetic,the original data set can be approximated as follows:

f x,t 9

N

i 1

a i c i

3.3.Cross-validation

The cross-validation process is needed to ensure that the right number of neurons is used in the network.Over-®tting is very common when using neural networks because one wants to make the error as small as possible.However,this may result in higher order®tting than is actually needed.Therefore,a mechanism to ensure over®tting does not happen should be devised.The cross-validation process involves the prediction of certain patterns not used in the training process.If the prediction is reasonable,the network is retained;otherwise,it is rejected.The search for the best network continues by increasing or decreasing the number of neurons until a satisfactory one is obtained.

4.Results

An experimental setup was built for a bent-axis piston pump.The purpose of the setup is to study the pump and minimize its power losses at high pressures.Our initial work is to derive a model to predict the dynamic behavior for the current design.There are several modeling schemes that can be used to derive a theoretical model for the pump.However,most of these schemes are simplistic and may not capture all aspects of the dynamic behavior of the pump such as nonlinearities.For that reason,neural networks are used here to attempt to capture most of the dynamic phenomena describing the system.Several neural networks were designed for the pump system following the procedure described in the previous section using a Pentium PC and Matlab.1The network that best describes the pump system has two hidden layers with ®ve neurons in each of the hidden layers.Most of the other designs were rejected because they failed in the cross-validation process.The neural network modeling technique predicted the steady-state behavior of the pump accurately for several pump speeds and pressure settings.The error between the experimental and the theoretical values does not exceed 2%(see Figs.7and 8).The pressure P was predicted accurately especially after time t =0.2s.At the start,the data was very noisy which can cause problems in most common ®tting techniques.This type of problem can be avoided in neural networks by using the proper type of transfer functions and ensuring that over-®tting does not occur.Despite the noisy nature of the training data,the designed neural network predicted the pressure in an acceptable manner.Figs.10and 12show a good agreement between the neural network prediction and

the Fig.7.Steady-state ¯ow rate for di erent set points and a pump speed of 1000rpm:experimental (Â)and NN prediction (dash-dotted)for set pressure of 75bar,experimental (w )and NN prediction (solid)for set pressure of 125bar,experimental (+)and NN prediction (dashed)for set pressure of 160bar.

1Matlab is software packaged and marketed by the Mathworks Inc.

experimental data.At the onset of the experiment;i.e.time less than 0.25s,the data looks very noisy and very hard to ®t.The designed network approximated the data in that region without ®tting the noise.Similarly,Fig.9shows how well the neural network model predicts the pressure P 2.The errors between the predicted and the experimental values are less than 7%.This is a

good Fig.8.Steady-state ¯ow rate for di erent pump speeds and set pressure of 75bar:experimental (Â)and NN prediction (dash-dotted)for 1000rpm,experimental (+)and NN prediction (dashed)for 800rpm,and experimental (w )and NN prediction (solid)for 550

rpm.

Fig.9.Prediction of the pressure P (ori®ce diameter=2.5mm):experimental (w )and NN prediction (solid)for pump speed of 1000rpm,experimental (+)and NN prediction for pump speed of 800rpm.

pump speed of1000rpm,experimental(+)and NN prediction(dotted)for pump speed of550rpm.

pump speed of1000rpm,experimental(+)and NN prediction(dotted)for pump speed of800rpm.

indication that the neural network modeling technique is a viable tool in modeling complicated systems such as the bent-axis pump.

5.Conclusion

A modeling technique,neural networks,has been used to predict the steady-state and dynamic behavior of a bent-axis piston pump.Experimental data was collected from an experimental setup to train the network.The resultant model predicted the pressures accurately.Therefore,neural networks has a lot of potential in modeling complicated systems such as the bent-axis piston pump.

Appendix.The gradient descent method

The gradient descent method is commonly used to update the weights during the training phase.Let us assume that the network is composed of n layers and the input vector to the network has m components.The output from neuron x in layer k is calculated as follows:a xk F k

2 N k À1i 1

w ixk a i k À1 b xk 3x 1,2,F F F ,N k and k 1,2,F F F ,n 1 where for k =

Fig.12.Prediction of the pressure P 3(pump speed=800rpm and ori®ce diameter=2.5mm):experimental (w )and NN prediction (solid).

N 0 m and a i 0 I i 2 The output of the output layer can be obtained by setting k=n in Eq.(1).The sum squared error (sse )of the network is given by:

sse P p 1 N n j 1À

d j Àa jn Á2p 3

where d j and a jn are the desired and actual outputs from neuron j in the output layer n ,respectively.The subscript p represents a speci®c input vector.The goal of the training procedure is to obtain a suitable set of weights that leads to a minimum sse .The gradient descent of layer k is given by: d n Àk p D F n Àk p ÀW T n Àk Áp d n Àk p k 1,2,F F F ,n À1

4 where T represents the transpose of the matrix.The neuron gradient matrix for an input vector p is given by: D F k p diag d F k 1k ,d F k 2k ,F F F ,d F k N k k

5 and the corresponding weight matrix is given by: W k p P T T R w 11w 12ÁÁÁw 1N k w 21w 22ÁÁÁw 2N k **ÁÁÁ*w N k 1w N k 2ÁÁÁw N k N k Q U U S p

6

The values of the gradient descent for the output layer is given by:

d p n D F n p z n p

7 where

z n p  d 1Àa 1n , d 2Àa 2n ,F F F ,Àd N n Àa N n n ÁÃT p 8 Following the procedure described above,we can calculate the gradient descents of all the layers one by one starting from the output layer.When all the gradient descents for all the layers are calculated,the adjustments for the weights are calculated using the following update rule:ÀD w ijk Áp Z Àd jk a ik Áp 9 The constant Z is known as the learning rate.The actual weights can be updated as follows: w ijk p w ijk p À1

ÀD w ijk Áp 10

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