## Relationship to other projects/themes

WP2.2 was originally planned to focus on impact dynamics and this has now been refocused onto friction, and so has connections to WP4.4 on non-smooth mechanics. In fact WP 4.4 also has impact included, and the work on anti-optimisation done under this work package was based on a impact experiment. There are also links to WP1.2 on predictive modelling with uncertainty.

## Aims

To model friction effects in nonlinear structural dynamics.

## Progress to date

Dr Alessandro Cabboi has been working on the development of a rig to measure dynamic friction forces. Based on the concept of a pin-on-disc machine, controlled random fluctuations in sliding speed are applied to the pin using an actuator, and the resulting pin motion and fluctuations in friction force are monitored and used to construct a new type of frictional frequency response estimated over a range of mid-high frequencies (up to 4 kHz).

This frictional frequency response function is the transfer function computed using the dynamic sliding velocity as input and the measured force as output. This approach has been used to measure the frequency response function for the sliding friction of a nylon pin and a rotating glass disc under various conditions, and to correlate the results with theoretical predictions (see work package 4.4).

The experimental apparatus used for these tests is shown in *Figure 1*. Note that this rig is also project demonstrator 2. The test method has been shown to give accurately repeatable results, which provide a kind of ‘fingerprint’ of a sliding interface that can shed new light on the nature of sliding friction; the measurement has the potential to discriminate between rival proposed theoretical models for dynamic friction. This possibility is currently being tackled in close collaboration with Dr. Thibaut Putelat (University of Bristol) via WP 4.4. An extended rate-and-state model which includes one state variable and an additional tangential contact stiffness has been proposed, and is showing very promising agreement with the measured results.

Preliminary results from fitting the measurements reveal that the proposed model can faithfully mirror the observed phase and amplitude variation of the measurements, for a range of sliding velocities and normal forces. Full details are given in WP 4.4.

Sample results from the test rig are shown in *Figure 2*. This shows the amplitude of the measured transfer function plotted against frequency, for six tests with sliding speeds that vary from 2mm/s to 10mm/s, all at the same constant normal force of 30N. The fact that these dynamic results are highly repeatable is demonstrated by two test results which were carried out using the same sliding speed of 2mm/s, but at opposite ends of a period of testing lasting several hours.

The results for these two tests are at the top of the plot, and can be seen to overlay each other almost exactly. This agreement is the more remarkable because the mean coefficient of friction, measured concurrently with these tests, was found to have increased by almost a factor of 2 over the same time period, probably because of a gradual rise in temperature caused by frictional heating.

The other results in *Figure 2* show the effect of progressively increasing the sliding speed, in intervals of 2mm/s up to a maximum value of 10mm/s. Milestone MS11 of this theme, due in month 33, required that the experimental dynamic friction measurements be fully operational and validated. Based on the results so far this milestone will easily be completed on time, with [7] already submitted. Furthermore these recent results open new avenues for experimental work, and also for modelling and for applications through simulation which will interact with the work of colleagues in the ENL project.

Natural possibilities for further experimental work include:

- testing of additional material combinations, and a wider range of test conditions, to probe how far the predictive power of the new theoretical model can be pushed
- use of the same test rig to perform alternative dynamic friction tests (such as velocity-jump tests as commonly used in the world of geophysics)
- further development of the test rig to allow identification of the full 2 × 2 matrix of frictional responses by bringing the normal degree of freedom fully into play.

Such tests will also feed the modelling agenda, providing the identification of parameters in the new sliding friction model. It will also allow the model to be tested under different dynamic conditions and with different materials and contact topography (with Dr. Thibaut Putelat). Furthermore, the new friction model will be explored in the context of the prediction of the stability threshold and the waveforms of fully-developed stick-slip vibration in applications such as brake squeal (with Dr Tore Butlin and Mr Andrew McKay).

It should be noted that for any industrial application where the stability threshold for friction-driven vibration is sought by linearised analysis, the frictional frequency response typified by *Figure 2* provides the input data required by any theoretical model, and the test rig developed here may in time be developed into a commercial tribometer for industrial testing. In the light of the measurements reported here, it is little surprise that attempts to predict brake squeal, for example, without having access to this input data have shown very little success.

Dr Tore Butlin’s work on anti-optimisation represents a novel approach to modelling uncertainty within complex nonlinear systems. The method exploits the fact that, for many systems, nonlinearities are spatially localised. The worst-case response is sought subject to constraints that are chosen to describe general features of the nonlinearity. Proof-of-concept studies have been carried out for simple idealised systems (mass on spring [1]), complex engineering structures (a turbine blade [2]), and laboratory experiments (impacting beam [3]). In recent developments, analytic upper bound responses have been derived for some general classes of nonlinearity. These predictions have been experimentally validated using results from the impacting beam laboratory experiment [3].