## Abstract

Under-platform dampers (UPDs) are traditionally used in aircraft engines to reduce the risk of high cycle fatigue. By introducing friction in the system, vibrations at resonance are damped. However, UDPs are also the source of nonlinear behaviours making the analysis and the design of such components complex. The shape of such friction dampers has a substantial impact on the damping performances, and topology optimisation is seldomly utilised—particularly for nonlinear structures. In the present work, we present a numerical approach to optimise the topology of friction dampers in order to minimise the vibration amplitude at a resonance peak. The proposed approach is based on the moving morphable components framework to parametrise the damper topology, and the efficient global optimisation algorithm is employed for the optimisation. The results demonstrate the relevance of such an approach for the optimisation of nonlinear vibrations in the presence of friction. New efficient damper geometries are identified in a few iterations of the algorithm, illustrating the efficiency of the approach. Results show that the most efficient geometry divides the vibration amplitude at resonance by 3, corresponds to a lower mass (80%) and a smaller frequency shift compared to the non-optimised case. More generally, the different geometries are analysed and tools for clustering are proposed. Different clusters are identified and compared. Thus, more general conclusions can be obtained. More specifically, the most efficient geometries correspond to geometries that reduce the mass of the damper and increase the length of the contact surface. Physically, it corresponds to a reduction of the initial normal contact pressure, which implies that the contact points enter stick/slip earlier, bringing more damping. The results show how topology optimisation can be employed for nonlinear vibrations to identify efficient layouts for components.

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## Acknowledgements

E. Denimal and L. Salles have received funding from Rolls-Royce and the EPSRC under the Prosperity Partnership Grant CornerStone (EP/R004951/1). L. Renson has received funding from the Royal Academy of Engineering (RF1516/15/11). E. Denimal has received funding from the City of Rennes. Rolls-Royce, the EPSRC, the Royal Academy of Engineering and the City of Rennes are gratefully acknowledged.

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## Contact modelling

### Contact modelling

One contact element is represented in Fig. 14 (Petrov and Ewins 2003). The normal contact force \(f_n\) at one point is given by

and the tangential contact force \(f_t\) by

where \(\mu\) is the friction coefficient, \(k_n\) is the normal contact stiffness, \(k_t\) is the tangential contact stiffness, \(N_0\) is the normal pre-load, \(x_0 = x(\tau _{stick})\) the displacement at the beginning of the stick state and \(f_t^0\) the tangential force at the beginning of the stick state. A complete description of the contact formulation can be found in Petrov and Ewins (2003).

The initial contact pressure \(\sigma _0\) at the contact interfaces is obtained from the equations of equilibrium of the centrifugal loading \(C_F\), the friction forces and the normal forces applied to the UPD without accounting for the elastic deformation of the damper and blade platforms (Petrov and Ewins 2007). It is given by

where *A* is the contact area on each side of the damper and \(\alpha\) is the damper angle (angle formed by the two contact surfaces). The centrifugal loading \(C_F\) is given by \(C_F = m_{damper}\times R \times \omega _r^2,\) where \(m_{damper}\) is the mass of the damper, *R* the radius and \(\omega\) the rotational speed. In the present case, for a full damper, the mass \(m_{damper}\) is equal to 3.42 g, the radius is taken as the distance between the gravity centre of the damper and the bottom of the base, and \(\omega _r = 1402\) rad/s. In the case of a full damper, it gives \(\sigma _0=9.8987\) N/mm. The normal contact stiffness \(k_n\) is set to 20000N/mm and the tangential contact stiffness \(k_t\) is assumed to be equal to \(k_n\) (Sextro 2007).

It is worth emphasising here that the damper properties that are directly related to its geometry will be updated during the optimisation process, i.e. its mass, the radius and the contact area. This has a direct impact on the centrifugal loading and the normal pressure.

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Denimal, E., Renson, L., Wong, C. *et al.* Topology optimisation of friction under-platform dampers using moving morphable components and the efficient global optimization algorithm.
*Struct Multidisc Optim* **65**, 56 (2022). https://doi.org/10.1007/s00158-021-03158-w

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DOI: https://doi.org/10.1007/s00158-021-03158-w

### Keywords

- Friction damping
- Topology optimisation
- Nonlinear vibrations
- Kriging
- Efficient global optimisation
- Moving morphable components