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Measurement of positive displacement pump flow ripple and impedance D N Johnston, BSc, PhD and J E Drew, BEng School of Mechanical Engineering, University of Bath

The secondary source method forms the British Standard for pumpfluid-borne noise testing. This is a powerful technique but requires care in order to produce accurate results. This paper describes practical aspects for implementing the method. The requirements for the test rig, data acquisition system and analysis are detailed. The British Standard specifies that either mathematical modelling or linear interpolation is used on the source impedance measurements. A method for smoothing the impedance results is described in this paper, which is shown to give more repeatable results than linear interpolation. Some physically realistic mathematical models of pump impedance are described, and their use in determining the interna1,flow ripple discussed. K e y words: pressure ripple, fluid-borne noise, pump flow ripple, source impedance, positive displacement pump

NOTATION

pipe internal cross-sectional area polynomial coefficients effective bulk modulus speed of sound effective capacitance of pump discharge passageway sum of squares error frequency frequency bandwidth for weighting function frequency of ith harmonic

4-1 Bessel functions of the first kind of order 0 and 1 effective length of pump discharge passageway effective inductance of pump discharge passageway number of sampled data points complex harmonic amplitude of pressure ripple inside pump complex harmonic amplitude of forward travelling pressure wave complex harmonic amplitude of pressure ripple at pump exit complex harmonic amplitude of reverse travelling pressure wave complex harmonic amplitude of pressure ripple at distance x complex harmonic amplitude of flow ripple inside pump complex harmonic amplitude of flow ripple at pump exit complex harmonic amplitude of source flow ripple of pump complex harmonic amplitude of flow ripple at distance x effective resistance of pump discharge passageway The MS was received on 20 March 1995 and was accepted for publication on 24 October 199s.

101095 @ IMechE 1996

term in transfer matrix time series windowing function frequency weighting function distance along pipe measured data point curve-fitting function source impedance of pump pipe characteristic impedance pump discharge passageway characteristic impedance 01

Y V

t P Ps 0

0,

non-dimensional damping coefficient wave propagation coefficient kinematic viscosity wave profile correction factor fluid density source reflection coefficient angular frequency natural frequency 1 INTRODUCTION

Positive displacement pumps are usually the main source of noise in hydraulic systems. This is predominantly a result of the inherent flow ripple caused by the pumping mechanism. The flow ripple interacts with the circuit in a complex manner to produce pressure ripple or fluid-borne noise. This in turn causes vibration and airborne noise. It is useful to be able to measure the pump fluid-borne noise characteristics. However, the pressure ripple is strongly dependent on the circuit and a simple pressure ripple measurement is not adequate (1, 2). A measure that is independent of the circuit is required. It is common to use linearized frequency domain techniques for the analysis of fluid-borne noise. Pressure and flow ripple can be represented as amplitude and phase spectra, and impedance is defined as the ratio of pressure ripple to flow ripple. This is analogous to electrical circuit theory, with pressure ripple equivalent to voltage and flow ripple equivalent to current. A pump or motor is commonly represented as a source flow Roc Instn Mech Engrs Vol210

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ripple Qs and source impedance Z , in parallel. This is equivalent to the Norton model which is used for electrical sources. The source flow ripple depends on the dynamic pumping behaviour and the source impedance depends on the dynamic characteristics of the pump discharge passageway. It is generally accepted that Qs and Z , are independent of the circuit characteristics, Although nonlinearities may mean that this is not strictly true, it is normally a close approximation. If Qs and Zs can be measured, they can provide useful information about the dynamic behaviour and noise-generating potential of the pump or motor (3). In addition, the fluid-borne noise generated in a connected circuit can be predicted if the circuit characteristics are known. The secondary source method has been adopted as the British Standard (4) for measuring pump fluid-borne noise. This requires some sophisticated data acquisition and analysis in order to obtain best results. This paper describes how the method may be applied in practice. Some improvements to the techniques used in the British Standard are suggested. Analysis techniques are presented for referring the pump flow ripple to the region where it is generated inside the pump. This internal flow ripple can provide important information about the pumping behaviour. The source flow ripple and source impedance, and the techniques for measuring them, are equally applicable to the suction and discharge ports of positive displacement pumps and motors. This paper will concentrate solely on the discharge characteristics of pumps, as this is the most common area of study. Most of the techniques presented in this paper are also applicable to the measurement of the impedance or transfer matrix characteristics of other hydraulic components (5). 2 MEASUREMENT TECHNIQUES

Using the Norton model, the flow ripple at the pump discharge port is defined by the equation PO

(1)

Qo = Qs --

Z S

The pressure ripple Po and flow ripple Qo at the pump discharge port can be evaluated from measurements of pressure ripple at points along a pipe, as described in Section 2.2 and the Appendix. However, it is necessary to take at least two sets of measurements, altering the

ratio of Po to Qo, to determine Qs and 2,. One way to achieve this is to change the impedance characteristics of the circuit, by altering the length of pipe (I, 6) or changing the termination impedance (7, 8). The accuracy of these techniques may be limited by two main factors. The pump flow ripple must not vary significantly between measurements when the circuit configuration is changed. There must also be sufficient change to the entry impedance of the circuit. This may be diffcult to achieve over the whole of the required frequency range. An alternative technique is described in Section 2.1. 2.1 The secondary source method In the secondary source method (2,9), the calculation of the source impedance Z , is separated from the calculation of the source flow ripple Q s . This is done by using a secondary source of pressure ripple, situated downstream of the pump as shown in Fig. 1. The secondary source is operated at a different speed to the test pump, and the harmonic components of the secondary source are measured. At these frequencies, provided that they do not coincide with harmonic frequencies of the test pump and that spectral leakage is negligible, the pump source flow ripple Qs can be assumed to be zero. The source impedance can be evaluated using the following equation:

zS -- _ - P O Qo

The pump flow ripple is determined by measuring the harmonic components of the test pump, with the secondary source not operational. The source flow ripple Qs can be determined from equation (l), provided that 2, is known at that frequency. There are two main difficulties in applying the secondary source technique. The first difficulty is to minimize interference from the harmonic components of the test pump while measuring the source impedance. Suitable signal analysis techniques are described in Sections 3.1 and 3.2. The second difficulty is to evaluate the source impedance accurately at the test pump harmonic frequencies, given that measurements of the source impedance are taken at different frequencies and may be subject to experimental scatter. The British Standard (4) specifies two alternative methods for determining the impedance at the required frequencies. These are linear inter-

Pressure transducers

Test pump

Secondary source

Small-diameter Loading valve

Fig. 1 Hydraulic circuit for secondary source technique Part I : Journal of Systems and Control Engineering

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MEASUREMENT OF POSITIVE DISPLACEMENT PUMP FLOW RIPPLE AND IMPEDANCE

polation between adjacent impedance measurements and fitting a mathematical model to the set of impedance measurements using a least squares optimization technique. The experimental scatter in the impedance results can cause errors when using linear interpolation. An alternative smoothing technique to reduce the effect of scatter is described in Section 4. Mathematical modelling of the source impedance also minimizes the effects of experimental scatter. Use of a physically realistic model can facilitate the calculation of the internal flow ripple. Mathematical modelling techniques based on simplified physical representations of the discharge passageway are described in Section 5 of this paper. 2.2 Test rig and secondary source Figure 1 shows the circuit diagram for the secondary source technique. The discharge port of the test pump connects to a length of uniform metal pipeline in which three miniature pressure transducers are mounted. Piezoelectric transducers are best as they only respond to pressure changes, so the mean pressure does not result in a large d.c. offset on the output. It is important that the length of line between the pump and the furthest transducer has a uniform internal diameter. However, it is acceptable to have small changes of diameter in the coupling between the pump and the pipe as these can be allowed for in the analysis. The transducers are mounted with their diaphragms flush with the inner surface of the pipe. To minimize the effect of pipe wall compliance, a thick-walled pipe is recommended. The diameter of the pipeline can affect the accuracy of the results, and its internal diameter should be close to the pump port diameter. Between 80 and 130 per cent of the pump port diameter is recommended. Downstream of the transducers are a pressure gauge, loading value, relief valve and secondary source. It is possible to perform the analysis using just two pressure transducers. However, problems can occur if the transducers are spaced at distances close to integer multiples of half a wavelength. At such conditions the analysis becomes ill-conditioned and large errors can result. This can be avoided by using three unequally spaced pressure transducers. Suitable transducer positions are given in the British Standard (4). The use of three transducers has the added advantage of enabling the accurate measurement of the speed of sound (10). Provided that the density is known, the effective bulk modulus can then be found. The accuracy of impedance measurements depends on the strength and stability of the harmonics of pressure ripple produced by the secondary source. Suitable devices for use as the secondary source are described by Edge and Johnston (9). The results presented later in this paper were obtained using a rotary valve which opens for very short durations during its rotation, allowing a bleed-off from the system. A schematic diagram of this device is shown in Fig. 2. Four pulsas are produced per revolution. The flow pulses produced by it are of extremely short duration, so the yressuie ripple produced in the system has a very wide frequency bandwidth, with the significant harmonics up to 10 kHz. This device was driven by a small variable-speed electric motor. Q IMechE 1996

67

-

Port A

\I,/-

’

1PortB

Fig. 2 Rotary valve pulse generator

Normally one port of the rotary valve is connected to the measurement line and one to the return line. The best signals are produced with a high pressure difference across the ports. When testing components at a low mean pressure, it is often more effective to apply a higher pressure at the other port. It is then necessary to avoid cavitation in the pulsed jet through the rotary valve, as the compressibility of the gas bubbles has been found to reduce the harmonic content of the pressure ripple. Cavitation occurs if the pressure ratio between the high-pressure and low-pressure ports is too large. The circuit is arranged to maximize the transmission of pressure ripple from the secondary source to the test pump. Side branches can act as resonators and attenuate the pressure ripple over particular frequency bands, and these need to be avoided. The diameter from the secondary source to the pressure transducers should be kept fairly constant and the volume of fluid in side branches minimized. The pipework associated with the loading valve and relief valve can produce undesirable resonances. Best results have been obtained by fitting these components to the end of a length of smalldiameter pipe. The relatively high entry impedance of this pipe has the effect of isolating the loading valve and relief valve impedances from the main circuit. The diameter of this pipe can be chosen to give the highest acceptable pressure drop along it. The pressure gauge can also cause resonances and should be fitted with a ‘snubber’ valve, which should be fitted as close as possible to the test pipe. 3 DATA ACQUISITION AND DIGITAL SIGNAL ANALYSIS

The test procedure has been implemented using a readily available high-speed data acquisition card connected to a personal computer. The data acquisition card employed has a 12 bit analogue to digital converter (ADC) with a minimum sample interval of 1 ps per channel, and up to 8 multiplexed input channels. It also has on-board memory enabling it to store 1048 576 samples. Signal analysis is performed using the computer. High-performance digital signal processor systems or dedicated instruments could be used, but with an increase in cost. Roc Iostn M a l i Engrs Vbl 210

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An optical sensor is used to give one pulse per revolution of the secondary source. This is used as a freqency and phase reference for impedance measurement. A similar signal is obtained from the test pump for use in measurement of its flow ripple. The three pressure ripple signals and one reference signal are recorded using the personal computer and high-speed data acquisition card. Unless the data acquisition card has a separate sample-and-hold device for each channel, there will be a delay of one sample period between the measurements on consecutive channels. This can readily be compensated for in the analysis by applying a small phase correction to the harmonic values.

3.1 Frequency resolution and spectral leakage Fourier analysis is performed on the measured signals. The test method requires very high frequency resolution and noise rejection, while any variations in the pump or secondary source speed need to be tracked. When performing data acquisition for source impedance evaluation, it is necessary to measure the pressure ripple produced by the secondary source. However, the test pump is normally operational and the pressure ripple from this is superimposed upon that from the secondary source. In order to measure the pressure ripple components produced by the secondary source, the harmonic frequencies must be different to those produced by the test pump. In addition, the frequency resolution of the Fourier transform must be maximized and spectral leakage minimized. This is achieved by using very long sampling sequences and applying a window to the time domain samples. Various types of window are suitable; the one used by the authors is a four-term Blackman-Harris window (11).This produces extremely small sidelobes (-92 dB) and is relatively simple to compute using the following weighting function for the nth data point:

(*r)

~ ( n=) 0.35875 - 0.48829 cos -

+ 0.14128):cors

- 0.01168 c o s ( T )

(3)

The duration of the window is limited by the amount of memory available on the data acquisition card, and also depends on the sample rate. Using 1 million samples with a typical sample rate of 60 kHz on four channels, a window of approximately 4 seconds’ duration is available, giving a 40 dB rejection bandwidth of approximately 1.25 Hz. Rejection of unwanted signal components can be improved further by taking several measurement windows in sequence and ensemble averaging the time domain data. It is not necessary for successive windows to be contiguous, and there is a delay between measurement windows as the data are downloaded into the computer’s memory and processed. 3.2 Order analysis The greatest accuracy is obtained if the discrete Fourier transform frequency bands correspond exactly with the harmonic frequencies or ‘orders’ of the periodic signal. This can be done most readily by sampling at a rate governed by an incremental encoder on the pump or Part I: Journal of Systems and Control Engineering

secondary source drive shaft. The discrete Fourier transformation can then be performed directly on the measured data. Using this order analysis, pump or secondary source speed variations are tracked automatically. Phase distortion and ‘smearing’ of harmonic peaks are thus minimized. However, it is important that such speed variations are kept to a minimum otherwise the accuracy of the subsequent pressure wave analysis will be impaired. In many cases it is difficult to connect an encoder to the pump or secondary source shaft. Also, additional instrumentation is needed to measure the speed of the shaft. A convenient alternative technique has been employed in the secondary source method which only requires a reference pulse signal once per revolution. The pressure ripple signals and reference signal are sampled at a fixed clock frequency. The samples between successive reference pulses are then mapped using linear interpolation on to a vector of 2” points (typically 256, 512 or 1024 points have been used). The vector represents the pressure ripple over one pump or secondary source cycle. In general the measurements are performed over several pump or secondary source cycles. The pressure data from each cycle are overlaid on to the same vector, which represents the cumulative sum of data from all cycles. A fast Fourier transform,is performed on this vector. Negligible loss of accuracy has been found using this method. The speed, and any speed variations, can be measured directly from the sampled data to a very high accuracy. Aliasing can occur if the sample rate is too low. It can also occur if the data are mapped on to insufficient interpolated points over one revolution of the pump. In the authors’ experience, sample rates of about 40-60 kHz, with 1024 interpolated points, have generally been found to give good results and anti-aliasing filters have not been necessary. The effects of aliasing can be greatly reduced by using long sampling sequences to minimize spectral leakage, provided that the aliased components do not coincide in frequency with the measured harmonics. If there is doubt as to whether aliasing is occurring, checks should be performed with higher sample rates and more interpolated points. 4 INTERPOLATION AND SMOOTHING OF SOURCE IMPEDANCE

An impedance smoothing technique that has been found to give good results uses a weighted linear regression technique to fit a function to the measured data over a localized frequency range. This is done by minimizing E where, for n measured data points, (4) W is a weighting function to emphasize points close to the required frequencyf. The following function, equivalent to a Gaussian distribution, has been used: 6 = exp( - x:) (5) where

fi -f x. = ’

SBAND

Y is taken to be a second-order polynomial with complex coefficients as follows (though other functions @ IMechE 1996

MEASUREMENT OF POSITIVE DISPLACEMENT PUMP FLOW RIPPLE AND IMPEDANCE

69

could be used): = a,

+ alxi + a2x?

(7)

The values of a,, a, and a2 are adjusted to minimize the error E in equation (4) by solving the matrix equation:

I

n

n

n

II

I OJ

0

4

2000

4Ooo

8000

6Ooo

Frequency Hz

I I=’ I I

i=l

The smoothed result at the required frequencyfis Y(0). This process is done at each frequency at which the smoothed value is required. An appropriate value for the weighting bandwidth fBAND needs to be chosen. This is a compromise between minimizing the effects of experimental scatter and following the form of the measured points. The source impedance 2, tends to exhibit resonances and anti-resonances which result in a wide dynamic range and sudden changes in the amplitude gradient and the phase. Performing the smoothing on such a function is not ideal; the regions of high amplitude tend to dominate and the ‘sharpness’ of the resonances and anti-resonances will be lost. A more appropriate function is the source reflection coefficient, given by the equation L,

I90 8Ol

I

-90 -

\

-I80 O0

2000

4OOo

6Ooo

x

*

8000

Frequency

Hz

Fig. 3 Axial piston pump source reflection coefficient

two source flow ripple amplitude spectra. These were produced using the same flow ripple test data but two different sets of impedance results, set A being that shown in Fig. 4. Linear interpolation was used for the

-Lo

Ps = -

(9) 2s+z o This generally varies more smoothly than 2,. An example of p s measurements is shown in Fig. 3, for an axial piston pump of 32 cm3 capacity operating at 1500 r/min and 100 bar. Here the smoothing procedure is applied at a large number of centre frequencies = with the bandwidth for the weighting functionf,,,, 500 Hz. The smoothed curves model the measured points quite closely, without rounding off the peaks and troughs. Generally the magnitude of ps is less than 1, as energy is lost in the reflection. Figure 4 shows the corresponding source impedance results. The smoothed curve follows the measured points quite closely and maintains the sharpness of resonant and anti-resonant peaks, without following the scatter on each point. Although not shown here, repeated impedance measurements were taken under identical conditions to this. While the form of the impedance results was the same in each case, the scatter was somewhat different as it exhibited significant random variation. Nevertheless, the curves obtained using the above smoothing technique were very similar. These impedance results were used in order to assess the sensitivity of the source flow ripple to the value of source impedance, Z,, in equation (1). Figure 5a shows @ IMechE 1996

14

0

t

2000

4Ooo

6Ooo

8ooo

6Ooo

8000

Frequency Hz

‘*O

T

x X

-180

4

0

I

2000

4OoO

Frequency HZ

Fig. 4 Axial piston pump source impedance Proc Instn Mech Engrs Vol 210

D N JOHNSTON AND J

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o Set B

0.04

X

0.01 - 0.1

1000

0

2000 Frequency

I

4OOo

3000

Hz (a)

Source impedance interpolation

-

0.05

x SetA

0.04 - -

g3

0 . 0 ~ .