When trying to understand a complex system, it is quite useful to have available some simple models – not as solutions per se but as organizational principles for seeing through the voluminous numbers produced by the FE codes. This chapter is concerned with the construction of simple analytical models; it gathers together many of the simple models used throughout the previous chapters and tries to illustrate the approach to constructing these. Although the models are approximate, by basing them on sound mechanics principles, they are more likely to capture the essential features of a problem and thus have a wider range of application. The models discussed are shown in Figure 7.1.

The term “model” is widely used in many different contexts, but here we mean a representation of a physical system that may be used to predict the behavior of the system in some desired respect. The actual physical system for which the predictions are to be made is called the prototype.

There are two broad classes of models: physical models and mathematical models. The physical model resembles the prototype in appearance but is usually of a different size, may involve different materials, and frequently operates under loads, temperatures, and so on, that differ from those of the prototype. The mathematical model consists of one or more equations (and, more likely nowadays, a numerical FE model) that describe the behavior of the system of interest.

Structures are to be found in various shapes and sizes with various purposes and uses. These range from the human-made structures of bridges carrying traffic, buildings housing offices, airplanes carrying passengers, all the way down to the biologically made structures of cells and proteins carrying genetic information. Figure 1.1 shows some examples of human-made structures. Structural mechanics is concerned with the behavior of structures under the action of applied loads – their deformations and internal loads.

The primary function of any structure is to support and transfer externally applied loads. It is the task of structural analysis to determine two main quantities arising as the structure performs its role: internal loads (called stresses) and changes of shape (called deformations). It is necessary to determine the first in order to know whether the structure is capable of withstanding the applied loads because all materials can withstand only a finite level of stress. The second must be determined to ensure that excessive displacements do not occur – a building, blowing in the wind like a tree, would be very uncomfortable indeed even if it supported the loads and did not collapse.

Modern structural analysis is highly computer oriented. This book takes advantage of that to present QED, which is a learning environment that is simple to use but rich in depth. The QED program is a visual simulation tool for analysis.

Suppose the existence of a very powerful computer, so powerful that it can execute any given command such as build a cantilever beam, excite the beam with this force history, record the velocity histories, and the like. Suppose, further, that it cannot answer questions such as what is elasticity? why is resonance relevant to vibrations? how are vibration and stiffness related? Here then is the interesting question: With the aid of this powerful computer, how long would it take a novice to discover the law governing the vibration of structures? The answer, it would seem, is never, unless the novice is a modern Galileo.

Now suppose we add a feature to the computer; namely, access to a vast bibliographic database (in the spirit of Google or Wikipedia) that can respond to such library search commands as find every reference to vibrations, sort the find according to the type of structure, report only those citations that combine experiment with analysis, and so on.

There are two main sources of nonlinearities in the mechanics of solids and structures. The first is geometric in nature and arises from large deflections, large rotations, and large strains. The second arises from the material behavior and is typified by elastic-plastic and rubberlike materials. Contact problems are also nonlinear even when the contacting bodies themselves remain linear elastic. Figure 5.1 is an example of a long, thin cantilevered plate undergoing large deflections and rotations because of an end moment.

The explorations in this chapter look at aspects of each of these nonlinear behaviors. The first exploration considers the concept of stress and strain under large deformation conditions because they need to be refined relative to the ideas explored in Chapter 2. The second and third explorations consider nonlinear material (constitutive) behavior in the form of elastic-plastic and rubber elasticity, respectively. The presence of a nonlinearity can affect the fundamental behavior of phenomena; the fourth exploration shows the rich, complex behaviors of nonlinear vibrating systems. Within a nonlinear context, applied loads affect the stresses, which in turn affect the stiffness and thus the deformations; this in turn affects the stresses, leading to a complicated connection between the applied loads and the responses. The fifth exploration uses vibrations under gravity to illustrate this point. The final exploration considers the contact that is due to impact of various shaped bodies and the resulting contact force histories.

When a structure is disturbed from its static equilibrium position, a motion ensues. When the motion involves a cyclic exchange of kinetic and strain energy, the motion is called a vibration. When this occurs under zero loads, it is called a free or natural vibration; whereas if the only loads are dissipative, then it is called a damped vibration. An example of a damped vibration is shown plotted in Figure 3.1; observe that the amplitudes eventually decreases to zero, but oscillates as it does so.

The explorations in this chapter consider the linear vibrations of structures. The first exploration uses a simple pretensioned cable to introduce the two basic concepts in vibration analyses; namely, that of natural frequency and mode shape. The second exploration looks at the meaning of mode shape in complex thin-walled structures. The stiffness properties are affected not only by the elastic material properties but also by the level of stress; the third exploration looks at the effect of prestress on the vibration characteristics. A significant insight into linear dynamics can be gained by analyzing it in the frequency domain. The fourth exploration introduces DiSPtool as the tool to switch between the time and frequency domains. Generally, increasing the mass of a structure decreases the vibration frequencies; however, in the presence of gravity, the mass can increase or decrease the stiffness and thereby affect the vibrations differently.

The Need for Fluid Power

In applications for which large forces, torques, or both are required, often with a fast response time, it is inevitable that oil-hydraulic control systems will be called on. They may be used in environmentally difficult applications because the drive part can be designed with no electrical components, and often they are the only feasible means of obtaining the forces required, particularly for linear actuation. A particularly important feature is that they almost always have a more competitive power–weight ratio when compared with electrically actuated systems, and they are the inherent choice for mobile machines and plants. Fluid power systems also have the capability of being able to control several parameters, such as pressure, speed, and position, to a high degree of accuracy and at high power levels. The latest developments are now achieving position control to an accuracy expressed in micrometers and with high-water-content fluids. In practice, there are many exciting challenges facing the fluid power engineer, who now must preferably have skills in several of the following topics:

• Materials selection, water-based fluids, higher working pressures

• Fluid mechanics and thermodynamics studies

• Wear and lubrication

• The use of alternative fluids, given the environmental aspects of mineral oil, together with the extremely important issue of future supplies of mineral oil

• Energy efficiency

• Vibration and noise analysis

• Condition monitoring and fault diagnosis

• Component design, steady-state and dynamic

• Circuit design, steady-state and dynamic

• Machine design and its integrated hydraulics

• Sensor technologies

• Electrical–electromagnetic design

• Computer control techniques

• Signal processing and associated algorithms

• Modern control theory and artificial intelligence

Introduction to Basic Concepts, the Hydromechanical Actuator

The interconnection of components to form a closed-loop control system introduces new features, primarily the consideration of steady-state error and the speed of control. Steady-state error is a function of component design, such as the existence of a servovalve spool underlap, and the speed of control is a function of the way control is dynamically achieved combined with the dynamic characteristics of the system. It has been shown in Chapter 5 that fluid compressibility and load mass–inertia play the dominant part when characterizing system dynamics, and these aspects cannot be neglected when the dynamic stability of a closed-loop system is considered. Increasing system gain – for example, by increasing a servovalve servoamplifier current gain in a closed-loop servovalve–actuator position controller – will eventually lead to closed-loop instability. This will result in severe oscillations that could rapidly lead to component damage.

The hydromechanical actuator is one of the simplest forms of closed-loop control with applications reaching back to the very earliest days of industrial manufacturing using cast-iron components and low-pressure water as the working fluid. It incorporates a spool valve and an actuator – for example as shown in Fig. 6.1.

It will be seen that as the handle is moved to the right, the spool valve opens, allowing pressurized fluid to move the actuator in the same direction as the handle movement. In its basic form, it therefore acts in a manner similar to that of a servovalve–actuator system.

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.