Introduction

These studies represent a variety of mathematical and simulation solutions for a range of components and systems and also include much experimental testing with some novel measurement techniques and practical limitations. They are intended to bring together the various aspects of fluid power theory introduced in earlier chapters, but in a more comprehensive manner usually required for more complex systems studies involving the integration of components and control concepts.

Performance of an Axial Piston Pump Tilted Slipper with Grooves

Introduction

This study was undertaken by Bergada, Haynes, and Watton with experimental work in the author's Fluid Power Laboratory at Cardiff University as part of a comprehensive study on losses within an axial piston pump. It was concerned with a new analytical method based on the Reynolds equation of lubrication, with experimental validation, to evaluate the leakage and pressure distribution for an axial piston pump slipper, taking into account the effect of grooves.

The analytical work was developed by JM Bergada (UPC, Terrassa, Spain) with experimental work undertaken by JM Bergada and JM Haynes. Additional CFD analysis and test-rig design was undertaken by JM Haynes and J Watton. Further CFD results by R Worthing and J Watton are also presented in this overview.

The equations consider slipper spin and tilt and are extended to be used for a slipper with any number of grooves.

This book is aimed at undergraduate students as a second-year and beyond entry stage to fluid power. There is much material that will also appeal to technicians regarding the background to fluid power and the operation of components and systems. Fluid power is often considered a specialist subject but should not be so given that the same would not be said for electrical power. In fact, there are many applications for which fluid power control is the only possibility because of force/torque/power/environmental demands. In the past 20 years, a number of groups around the world have made significant steps forward in both the understanding and the application of theory and control, complementing the R&D activity undertaken within the manufacturing industry. Details of just one organization involving many participating fluid power centers around the world are available at www.fluid.power.net. I embarked on this book ostensibly as a replacement for my first book, Fluid Power Systems – Modelling, Simulation, Analog and Microcomputer Control, published by Prentice-Hall in 1989 and now out of print. However, the result is a much different book and perhaps not surprising, given the developments in fluid power in the past 20 years. Following many constructive comments by undergraduate students, friends in industry, and academic friends who still use my first book for teaching, it was clear that a new book was needed.

Fluid Types

The preferred working fluid for most applications is mineral oil, although in some applications there is a requirement for a water-based or synthetic fluid, mainly for reasons of fire hazards and increasingly for environmental considerations. The drive toward nonmineral oil fluids has seen a renewed attitude to pure water hydraulics together with the emergence of biodegradable and vegetable-based fluids. Fire-resistant fluids in use fall under the following classifications:

1. HFA 5/95 oil-in-water emulsion, typically 5% oil and 95% water

2. HFB 60/40 water-in-oil emulsion, typically 60% oil and 40% water

3. HFC 60/40 water-in-glycol emulsion, typically 60% glycol and 40% water

4. HFD synthetic fluid containing no water

5. HFE synthetic biodegradable fluid

The use of water-based fluids has implications for component material selection – for example, the use of stainless steel, plastics, and ceramics. In addition, serious consideration of fluid properties must also be given, particularly viscosity, which can be very high at low temperatures in some cases. Fluids are being continually developed, and the following information is intended to reflect the general trend and is not considered as definitive because this would require an overview of many suppliers from many countries around the world – for example, see www.shell.com.

Type HFA 5/95 oil-in-water emulsions are fire-resistant emulsions that exhibit enhanced stability, lubrication, and antiwear characteristics and have the following important aspects:

• They have much improved stability toward variations in temperature, pressure, shear, and bacterial attack.

• The performance limitations become obvious for systems operating well above 70 bar, reliability and efficiency often being sacrificed where fire resistance is of paramount importance.

• […]

Introduction

The preceding chapters considered the steady-state behavior of common fluid power elements and systems. In reality, fluid power systems handle significant moving masses, and the combination of this with fluid compressibility results in system dynamics that usually cannot be neglected. In addition, individual components such as PRVs require a finite time to accommodate flow-rate changes. This also applies, for example, to a servovalve that again requires a finite time to change its spool position in response to a change in applied current. The combination of these issues means that the design of both open-loop and closed-loop control systems must take into account these dynamic issues. In particular, a closed-loop control system will almost certainly become unstable as system gains are increased because of such dynamic effects. Instability can lead to disastrous consequences if severe pressure oscillations occur. Instability in axial piston motor speed control systems, for example, can result in severe repetitive lifting and impact of the pistons on the swash plate.

Consider the design of a servoactuator that forms one of four to be used to provide the “road” input to the wheels of a vehicle sitting on a rig commonly called a “four-poster.” Figure 5.1 shows one of the servoactuators and a block diagram of the position control system.

Determining the dynamic performance of the position control system only is relatively straightforward once the important dynamic features have been identified.

The effects of roundoff noise in control and signal processing systems, and in numerical computation have been described in detail in the previous chapters.

There is another kind of quantization that takes place in these systems however, which has not yet been discussed, and that is coefficient quantization.

The coefficients of an equation being implemented by computer must be represented according to a given numerical scale. The representation is of course done with a finite number of bits. The same would be true for the coefficients of a digital filter or for the gains of a control system.

If a coefficient can be perfectly represented by the allowed number of bits, there would be no error in the system implementation. If the coefficient required more bits than the allowed word length, then the coefficient would need to be rounded to the nearest number on the allowed number scale. The rounding of the coefficient would result in a change in the implementation and would cause an error in the computed result. This error is distinct from and independent of quantization noise introduced by roundoff in computation. Its effect is bias–like, rather than the PQN nature of roundoff, studied previously.

If a numerical equation is being implemented or simulated by computer, quantization of the coefficients causes the implementation of a slightly different equation.

The purpose of this chapter is to provide an introduction to the basics of statistical analysis, to discuss the ideas of probability density function (PDF), characteristic function (CF), and moments. Our goal is to show how the characteristic function can be used to obtain the PDF and moments of functions of statistically related variables. This subject is useful for the study of quantization noise.

PROBABILITY DENSITY FUNCTION

Figure 3.1(a) shows an ensemble of random time functions, sampled at time instant t = t1 as indicated by the vertical dashed line. Each of the samples is quantized in amplitude. A “histogram” is shown in Fig. 3.1(b). This is a “bar graph” indicating the relative frequency of the samples falling within the given quantum box. Each bar can be constructed to have an area equal to the probability of the signal falling within the corresponding quantum box at time t = t1. The sum of the areas must total to 1. The ensemble should have an arbitrarily large number of member functions. As such, the probability will be equal to the ratio of the number of “hits” in the given quantum box divided by the number of samples. If the quantum box size is made smaller and smaller, in the limit the histogram becomes fx(x), the probability density function (PDF) of x, sketched in Fig. 3.1(c).

The extremely fast rolloff of the characteristic function of Gaussian variables provides nearly perfect fulfillment of the quantization theorems under most circumstances, and allows easy approximation of the errors in Sheppard's corrections by the first terms of their series expression. However, for most other distributions, this is not the case.

As an example, let us study the behavior of the residual error of Sheppard's first correction in the case of a sinusoidal quantizer input of amplitude A.

Plots of the error are shown in Fig. G.1.

It can be observed that neither of the functions is smooth, that is, a high–order Fourier series is necessary for properly representing the residual error in Sheppard's first correction, R1(A, μ) with sufficient accuracy. The maxima and minima of R1(A, μ) obtained for each value of A by changing μ, exhibits oscillatory behavior. For some values of A, for example as A ≈ 1.43q or A ≈ 1.93q (marked by vertical dotted lines in Fig. G.1(b)), the residual error of Sheppard's correction remains quite small for any value of the mean, but the limits of the error jump apart rapidly for values of A even close to these. A conservative upper bound of the error is therefore as high as the peaks in Fig. G.1(b). One could use the envelope of the error function for this purpose.