Benefits & Applications

• Finite element modelling of ultrasonic/acoustic dynamics.
• Equivalent circuit modelling of ultrasonic/acoustic systems using SPICE.
• Hybrid finite element and SPICE for extreme aspect ratio samples, for example pipelines and risers.
• Theoretical analysis from first principles.
• Matlab and Mathcad numerical models for signal processing and DSP design.
• Custom Windows programming in C, C++, Visual Basic, Excel macros.
• Transducer design.

Cambridge Ultrasonics

Cambridge Ultrasonics/Cambridge Software offers mathematical modelling services related to ultrasonic engineering, signal processing and the physical sciences. We are one of the few independent, commercial organisations in the UK offering such a service. Some points to keep in mind when thinking about using a subcontractor for mathematical modelling are:

• Our goal is to give you high quality research expertise at a competitive rate and in a time-scale much shorter than a university.
• At Cambridge Ultrasonics we have staff with degrees in a variety of subjects including mathematics and physics up to PhD level.
• We have links with the Physics and Engineering departments in Cambridge University so we have access to a first rate pool of knowledge, experience and facilities.
• Cambridge Ultrasonics has staff with individual experience of over twenty-five years in ultrasound.
  We also specialise in collecting experimental images from our visualisation equipment and these can be used in some circumstances to validate numerical predictions.

Advantage of visualisation

A finite element program can be used to give time-lapse images of ultrasonic waves propagating in a test sample. Our schlieren visualisation equipment produces similar images but derived from experiments. Comparing the two allows a check to be made of the quality of the numerical predictions. This is a unique commercial capability and distinguishes our service from others.

Techniques

Cambridge Ultrasonics offers the following methods to the service of its clients:

• Finite element (FE) modelling.
• Schlieren visualisation for bench-marking FE results.
• Lumped circuit equivalent representation followed by SPICE analysis.
• Hybrid modelling using finite elements and SPICE.
• Matlab and Mathcad for numerical evaluation and DSP prototyping.
• Windows programming.
• Calculus from first principles.

Application - inspecting car bodies during manufacture

Some of our clients make cars, trucks and automobiles and have asked us to help develop better inspection methods for car bodies and components. The ultrasonic transducer is often important to the usefulness of the inspection method. We use FE modelling to prove the concept of a new transducer before embarking on the longer task of building and testing a new transducer. A FE model can generally be developed in a few hours or days to prove the concept of a transducer. Proving the operation by FE requires the computationally intensive task of predicting the propagation of elastic waves that propagate distances of generally many wavelengths. This can only be done by dividing the time of interest into many small time steps and using the final vlaue conditions from one solution as the initial values of the next solution. Adjusting material properties can make significant changes to performance and this is an area of particular interest in developing novel transducers. A common aim is to create wide-band transducers.

Application - ink-jet printers

Some of our clients make ink-jet printers with piezo-electric elements to modulate the pressure used to form ink droplets. They wanted to know the spatial distribution of the modulating pressure field in and around the printing head and chose finite element methods to help with the design. They also wanted experimental evidence to check the predictions. We built a transparent model of a print head to compare experimental images with predictions. Agreement was good so the client is confident that it has a sound method for designing new print-heads.

Application - volume fraction of mixed gases

How can you measure the quantities of two gases in a container without disturbing them? That was the question posed by one client. In this case we went back to first principles and thermodynamics to see how the speed of sound was affected by pressure and ratios of different gases. We derived equations of state and predicted the frequency of resonance of the gas in a Helmholtz chamber. We also helped by using Cambridge Ultrasonics’ resonance spectroscopy equipment for experimental results.

Application - oil and gas

Pipelines and risers are made of many lengths of tubing fixed together. The aspect ratio (length/diameter) is large and this creates problems for finite element modelling. One approach is to model a single section of tubing and apply Floquet’s theorem to get the response of many tubing sections fitted together but few FE programs support this approach. Another solution is to represent a section by equivalent lumped electrical circuit elements then analyse the full length circuit using a circuit analysis program (SPICE). This method is much faster both in terms of computational speed and creating a model but needs a good equivalent circuit. We have successfully used a hybrid approach where a finite element model is used to help create the equivalent circuit of one length of tubing.

Application - oil and gas

We have devised analytical solutions using thermodynamic methods to explain unexpected pressure fluctuations found in oil wells.

Application - crater volume estimation

Impact experiments on homogeneous metallic targets give smooth craters when solid spheres are used as projectiles. In one application it was important to know the volume of the crater to better than 10% accuracy. We developed a mathematical model of the crater profile from first principles and a multidimensional numerical minimising routine to fit the model to experimental data. We also developed a Moire method to put contours on the craters. The final error was 3% - well within target.

Application - time/frequency domain

We have used the Wigner transformation to look at ultrasonic dispersion in concrete. This is important because dispersion causes sharp pulses to lengthened, resulting in a loss of resolution.

Application - theory of random collisions

When solid particles strike a solid surface they can cause damage and remove material (erosion). It is important for materials scientists to understand the mechanisms of erosion so that they can choose materials to withstand erosive environments. Possible erosion mechanisms include: cutting, melting/ablation and deformation wear.

Much can be learnt from the development of erosion with time:

• There can be an incubation time before mass is lost from the eroding surface.
• There can be a period of mass-gain during incubation.
• After incubation there is commonly a period of linear erosion, where mass is lost steadily with time.
• If the flux of erosive particles is increased there can be threshold flux above which the rate of erosion increases significantly.
• If the flux of particles is increased to a high level then collisions can take place in the vicinity of the eroding surface that some think can protect the surface.
• It the test is conducted at different temperatures then the erosion rate is known to change.

We have investigated the theory of the statistical mechanics behind erosion in terms of impact-zones and predicted the erosion rate for various conditions. The arrival times of particles on a zone follow the Poisson distribution. Incubation times, flux dependence and temperature dependence all indicate the mechanisms acting. Sometimes choosing a material with a higher melting temperature will be sufficient to improve the lifetime of a component subject to erosion, sometimes finding a way to limit the flux of particles is sufficient.

We also theoretically investigated the effect of collisions by calculating the joint-probability of finding two particles close enough to collide. This is a more complex theoretical problem but we were able to show, for example, that collisions between eroding particles in the vicinity of the eroding surface do not reduce the number of particles striking the surface but do cause the stream of particles to become more divergent. We were able to quantify when a significant number of collisions would occur, in terms of eroding flux, and to define for the first time what information is needed to predict the flux - this is the information needed to define what flux means in the context of erosion.

Publications

• "Ultrasonics and Acoustics" Encyclopedia of Physical Sciences and Technology, Aceademic Press 2002.
• "Shadow moire contouring of impact craters" D.R.Andrews. Optical Eng 21,4(1982)650-4.
• "An analysis of solid particle erosion mechanisms" D.R.Andrews J.Phys.D:Appl Phys 14(1981)1979-91.
• "Temperature dependence of the impact response of copper: erosion by melting" D.R.Andrews, J.E.Field J.Phys.D:Appl Phys 15(1982)2357-2367.
• "Particle collisions in the vicinity of an eroding surface" D.R.Andrews, N.Horsfield. J.Phys.D:Appl Phys 16(1983)525-538.