Numerical Modelling at Rupert Taylor Ltd. Rupert Thornley-Taylor FIOA.

Numerical modelling at Rupert Taylor Ltd. has its origins in the late 1970s, when Rupert Taylor began writing 2 – dimensional ray-tracing programmes on a Hewlett Packard 9845, a cross between a giant programmable calculator and a PC (then unheard of). In the early 1980s, when the Ford Motor Company commissioned a study to find ways of quietening its then new York direct-injection diesel engine, Rupert Taylor's solution, subsequently patented by Ford, was to design a wave filter for insertion in the engine's fuel injector pipes. He began trying to find solutions to the differential equations governing non-linear wave propagation in liquids under very high pressure, and on failing then sought to model wave propagation by finite difference methods. This was all done on a Commodore PET computer with 16 kB of RAM, and was a little limited in scope. The slightly easier problem of linear wave propagation cropped up later with the design of the Channel Tunnel. Firstly, leaving finite differencing on one side for the time being. Taylor took to numerical modelling using transmission line theory, involving the superposition of forward and backward travelling waves in waveguides after applying reflection and transmission coefficients. This is a very neat technique which converges exactly with algebraic solutions, for example, of equations for the transmission loss of reactive attenuators. He made his first software sale to a French company in the form of a package for modelling low frequency transmission of sound from the tunnel space into the interior of shuttle wagons. This was followed by work carried out to study the likelihood or otherwise of pressure oscillations arising in the draught relief system of the Channel Tunnel itself. Interestingly, while the transmission line model used was intended to study oscillating flow, the results overlapped nicely with those of a simultaneous aerodynamic study carried out using a box model. Transmission line theory in acoustics can be found succinctly described in the first edition of C M Harris's Handbook of Noise Control, and is a powerful tool. A year or two later, having been impressed by a talk given by Manfred Heckl, Taylor discovered what remains to this day his secular bible, the late Ted Shulz's translation of Cremer and Heckl's book Structure-borne Sound. In here you will find transmission line theory applied to structure-borne sound. Soon Taylor was adapting the airborne transmission line model to a structural one, and using it for modelling the behaviour of the railway track on the then proposed Tsing Ma bridge, part of Lantau Fixed Crossing in Hong Kong. The problem with structural transmission line theory is that the phase velocity is dependent on frequency, and the model has to be run separately for each frequency of interest. At the time, a 33 MHz Intel 386 was the state of the art so that full spectrum modelling was slow, and the frequency limitations of transmission line models for structural wave propagation were the driving force behind a change of tack back to finite difference modelling. Finite difference modelling involves creating a representation of the structure or space to be modelled using a generally rectilinear grid. Each cell in the grid is assigned a shear modulus, mass, loss factor, displacement and velocity. For each time step (which may have to be as little as 20 microseconds to achieve a stable model), the shear and compressive stress, and the damping force are computed and the cell is accelerated accordingly. Its new displacement and velocity at the end of the time step are computed and the process repeated. For well known structures such as beams, correspondence with Timoshenko beam solutions is achieved, subject to small and quantifiable errors resulting from the finite difference process. The first finite difference model was ready for the design of railway tunnel vibration isolation systems in the CrossRail project. Validated by retrospective modelling of the Singapore MRT and the London Underground tunnel at Heathrow Terminal 4, the effects of all the design nuances of floating track slab and resilient baseplate track could be predicted. Further validation was provided through an exercise of a successful 'blind' prediction of levels of ground-borne noise inside an existing theatre affected by an existing underground railway, based on a measured spectrum of rail roughness in the actual tunnel. Meanwhile the floating track slab and resilient baseplate track of the Jubilee Line Extension had been designed and the FD model was employed to study their likely behaviour. Figure 1. Finite difference modelling of train, tunnel, surrounding soil, building and air above. Finite difference modelling of this kind is wholly deterministic, and the model can only be as accurate as the material properties supplied to it. Moreover, in a railway tunnel vibration is largely controlled by the magnitude of wheel and rail roughness, something which can only be taken as a given piece of input data. Measurements in the Channel Tunnel, where rail roughness measurement results were available and surface vibration measurements possible in a location where the tunnel is under land rather than sea gave added confidence in the level of accuracy achievable when properties such as soil shear modulus and loss factor are only known approximately. In any case where the future performance of a system such as an underground railway has to be predicted, a deterministic model can at least consider the effect of uncertainties in each of the parameters in the model, so that levels of confidence can be applied to the results. One of the most useful facilities that models of this kind offer is the ability to study intricate effects such as coincidences between bending wavelength in rails and floating track slabs and vehicle dimensions such as bogie wheelbase, with the train moving at any (not necessarily constant) speed. Many dynamic phenomena which ordinarily tend to take the railway world by surprise when the railway starts operating can be foreseen by operating the FD 'virtual' railway early in the design process. While modelling underground railways is one of the most valuable applications of FD modelling, in which the dynamic behaviour of the moving rail vehicle and its interaction with the track characteristics are all used to predict not only tunnel wall vibration but also propagation through the surrounding soil, more exotic applications beckon. Propagation of sound in air and modelling the acoustical behaviour of spaces, with audible output via a sound card to headphones is an extreme form of auralisation. Modelling a complete musical instrument even more so. Unlike the transmission line model which has to be run separately for each frequency when used in structural applications, the finite difference model copes with all frequencies at once, provided that the damping term can be represented as directly proportional to particle velocity. Damping which has a particular frequency dependence means that even the finite difference model has to be run separately for each frequency of interest, unless the damping mechanism can be understood well enough to build it into the FD equation. Figure 2. The Class 156 Diesel Multiple Unit enjoyed the benefit of numerical prediction of interior sound levels, followed by East Coast Main Line Coaching Stock, Class 456 Networkers and most recently Class 323 Electrical Multiple Units. While the modelling of structural vibration is by far the most complex application, which requires not only the handling of compression and shear, modelling of sound waves in air is no less complex when it comes to the behaviour of boundary surfaces. The dynamic behaviour of walls and panels is merely a special case of the structural model, but the resistive absorption of sound absorbent surfaces requires clever representation of the properties of the surface if not only the correct sound absorption coefficient is to be achieved, but also the correct phase relationship between incident and reflected sound is preserved. Figure 3. The design of the New Victoria Theatre, Stoke on Trent, was one of the earliest applications of Rupert Taylor's ray tracing software. The Geometric shape of the reflectors suspended above the stage area of the theatre-in-the-round was designed to maximise early reflections from each performer around the full 360 degree auditorium. Computational speed is obviously of great interest. The original HP calculator on which it all began had only ROM-based interpretive BASIC and ran at about the same speed as a Sinclair Spectrum which was also used for some ray tracing in the early days (subsequently replaced by a QL). While Z80 machine code is comparatively easy to write, Intel 80386 code is not. The big feature of the model is the size of the arrays used, far bigger than those conveniently handled by DOS compilers. IBM came to the rescue with the operating system they hoped would replace DOS, called OS/2. Version 1.0, which came out in 1986 and gave you no more than a prompt and CGA graphics, nevertheless had the pure joy of a flat memory model, virtual memory through disk swapping and true multitasking. Version 2.0 followed, and was a 32-bit operating system many years ahead of Windows 95 or NT. So while most people were struggling with DOS, Rupert Taylor was enjoying boundless memory size – bounded only by the capacity of the hardware which doubled each year. The rise of C++ language, which though totally unforgiving is almost as easy to write as the more advanced forms of BASIC, and very fast if the right compiler is used (Borland, C++ and OS/2 are excellent bedfellows) and the falling price of RAM, mean that complex structures such as the new combined station for the Jubilee Line and District Lines at Westminster, supporting phase 2 of the new parliamentary building over the top, can be modelled for as long as the passage of a complete train, in an overnight run. Real-time is a long way off (at least on a PC), but the output is printed to file which can then be converted into a .WAV file for replay through a sound card. Rupert Taylor is a consultancy practice which began life 31 years ago, when only a handful of present day practices around the world existed in the form of acoustical consultants. He claims to have been the first person to use the term 'noise consultant' in 1968, when it was a strangely off-beat profession. Now that noise consultants seem almost as plentiful as architects, moving into the more obscure corners of acoustical and dynamic modelling is like going back to 1968, into a field occupied by only a handful of people around the world. Rupert Taylor Ltd. is based at Uckfield, East Sussex and is a Member of the Association of Noise Consultants.