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March 28, 2007

The Knowledge

From the reviews section of The Geographical Journal, Vol. 124, No. 2 (Jun., 1958):

THE KNOWLEDGE OF LONDON (Cousland, 1958, 15s 6d) is a guide with a difference; intended for car-drivers, it gives directions to an enourmous number of places without recourse to maps. The author, Colin J. Hunt, calls this the “conversational” method of direction, and limits his maps to simplified outlines. The white-on-black design of these is somewhat dazzling at first sight, but they are very clear when the eye becomes used to them. DOROTHY MIDDLETON

It appears that the full title of this book is "Here it is" Knowledge of London rather than Guide to Learning the Knowledge of London, so clearly this isn't the actual Public Carriage Office blue book. This review doesn't suggest that it is intended to help would-be cabbies, although that would be plausible. (Or perhaps it was written by someone who didn't get his green badge but wanted to get something for his efforts.)

I'm now curious to know if it has Manor House Station to Gibson Square.

Posted by robin2 at 11:26 PM | Comments (0)

March 19, 2007

Differential Analysers

Differential equations

Introduction

A common form of mathematical modelling is to describe a system in terms of a set of (real-valued) variables which change with time. The state of the system at any particular time consists of the values for these variables (and so they are sometimes called "state variables"). And for each variable the way it changes can be given as a function of the state. So the model consists of a set of differential equations for the state variables; all differentiation being with respect to time.

A very simple example of this would be of an undamped spring-mass system. The state of this can be described by a displacement x of the weight from its rest position, and its velocity v. The equations would be of the form:
dx/dt = v
dv/dt = g - kx/m
[Here the system is characterised by a number of constants, in this example g, k and m; which are gravity, the stiffness of the spring, and the mass of the ball respectively.]

This gives a simple oscillation.

In biology there are models of population dynamics that work in a similar fashion. The Lotka-Volterra model is a simplistic account of a predator-prey relation: the size of a population of rabbits is given by R, that of foxes by F, and they are related by:

dR/dt = aR - bRF
dF/dt = ebRF - cF
[Where a, b, c & e are constants quantifying different aspects of rabbit and fox existence.]

This gives rise to a slightly difference cyclical behaviour. [See e.g. http://www.stolaf.edu/people/mckelvey/envision.dir/lotka-volt.html.]

Evaluation

For some models the equations can be solved analytically. But for the rest it is necessary that a model be 'run' in some fashion in order to observe its behaviour.

These days this would be done with some sort of step-wise simulation:

There are refinements, but this is basically how it works. In order to give good results time has to be divided up into small steps; and the smaller they are the more steps there will be in the simulation period. This makes for a large number of numerical calculations, and so this approach would have been impractical prior to the development of digital computers.

Analog computers

Introduction

Analog computers are different. An analog computer is a physical system that is contrived—through its construction—to be described by the same set of differential equations as the system that one is interested in, and so its behaviour is essentially the same as that of the original system (taking into account suitable conversion factors). Measuring the computer's behaviour gives the results of the simulation.

A differential analyser is an analog computer that is a mechanical system where the variables from the equations correspond to a set of axles, and the values of the variables correspond to the axles' speeds of rotation. The evaluation of expressions involving these variables is achieved through various forms of gearing.

[I assume that analog computer would have been made in the form of a 'kit' which would include components for performing particular arithmetic operations, which would then have to be linked together as appropriate for a particular job.]

It is obvious that the multiplication or division of an input variable by a constant can be represented by a simple gear of an appropriate ratio. There would have to have been other components to represent at least addition and subtraction too, and I would have thought there would be components for multiplying variables too, but I'm afraid I have no idea how these might have worked.

Integrator

From what I've just said it can be seen that the expression for the right-hand side of a differential equation can be implemented as a mechanism that takes as input the rotational speeds of the axles representing the state variables, and produces as output a rotational speed on another axle to represent the instantaneous change in the state variable indicated on the left-hand side of the equation. To make this actually work, the construction of the computer has to ensure that the speed of the axle representing a state variable really is controlled by the one that's supposed to represent its instantaneous change: in other words the mechanism has to ensure that the speed of the former is the integral of the latter.

A mechanical integrator features a disk that is rotated at a constant speed. On this there is a wheel that rotates on the axle representing the outputted state variable. The position of this wheel can be varied from one side of the drive disk to the other, controlled by the input axle that represents the expression of the differential. (I guess that it uses a rack-and-pinion to convert the input into a horizontal movement of the arm.) So a rotation on the input will move the wheel, changing the effective gear ratio between the drive disk and the wheel, and so changing the rotational speed on the output.
Torque Amplifier

One problem with the integrator as described is that the torque on the output is very low. The drive disk is smooth and the wheel doesn't have any teeth, so the output rotation cannot do very much without the wheel just slipping. The principle of mechanical integration was known to Lord Kelvin in the 19th century, but this sort of mechanical computer wasn't possible until Vannevar Bush invented the torque amplifier in the 20th century.

I did have a bit of mental block on this bit, but I think I've got it sussed now. The torque amplifier is normally described as working on the same principle as a ship's capstan. I've never seen a capstan. In fact I don't think I'd ever seen the word previously apart from a passing reference in Treasure Island. (The book, describing Billy Bones' voice, says that it “seemed to have been tuned and broken at the capstan bars.”)

My understanding of it is as follows. Suppose that I am trying to pull a heavy load with a rope, and I want to be able to control how fast it is pulled. Then suppose I have the rope looped over a rotating pillar (which is being powered by a motor). As I pull on the rope, the rope will tighten around the pillar, and the pillar will impart a force on the rope that is can be stronger than that from my pulling. So the load can be pulled with the motor doing most of the work. But it can't be pulled faster than I am pulling: if it did the rope would slacken around the pillar and so would not be powered by it any more. So the overall effect is that the force of my pulling is amplified, without changing its speed.

A torque amplifier works basically like this, although there are some complications. One is that a capstan only provides amplification in one direction, whereas the output axle of an integrator could be going clockwise or anticlockwise (corresponding to positive or negative values of the variable) and the torque has to be amplified which ever the direction. I get the impression that this is overcome by having two capstan mechanisms, one for each direction. (The rope for the one not in use at any particular time would be slack.)

It is clear that a capstan is a feedback system, with the force imparted by the rotating pillar depending on how much too fast or slow the rope is moving. This means that it will have its own response characteristics, which could be a potential problem. It is possible to imagine pulling the rope at a constant speed, but for the load to be pulled in a 'juddering' fashion, with the mechanism always over-compensating. Alternatively (if, I don't know, the rope was very spongy) there might be a considerable time lag between changes in how fast I pull and how fast the load is pulled. I imagine that in a torque amplifier the way of dealing with this is to err on the side of slow response over instability, but to ensure that the lag is insignificant in the time scale at which the overall machine works.

Posted by robin2 at 08:09 AM | Comments (0)

March 18, 2007

Oh that worked.

Posted by robin2 at 11:19 AM | Comments (0)