Nonsmooth modal analysis from elementary impact oscillators to turbomachines: Limitations and avenues

In mechanical engineering, nonlinear modes characterise the behaviour of nonlinear vibratory systems. In the current state-of-the-art, they are well defined for smooth nonlinear systems (of moderate size) of Ordinary Differential Equations governing the dynamics. They are defined as continua of periodic orbits forming two-dimensional invariant manifolds in the state space. This framework has lately been extended to nonsmooth mechanical systems involving impact dynamics. From the theoretical standpoint, strong mathematical results about existence, uniqueness and analytical or approximate equations of nonsmooth modes have recently been established on a linear spring-mass chain undergoing a Newton elastic impact law on one of its masses. From the industrial point of view, their application is not straightforward. This paper investigates the possibilities and the limitations of these tools with a practical end: the issue of blade-casing unilateral contact interactions in turbomachines. The main differences between a simple spring-mass chain and complex blade-casing models are investigated point by point: non-diagonal mass matrix, geometrical differences, scalability for thousands of degrees of freedom, convergence with respect to the number of dofs, stability and relationships with force and damped mechanical systems. It is found that the proposed formulation and corresponding solutions apply to very general spring-mass systems, involving non-diagonal mass matrices and non-tridiagonal stiffness matrices, opening avenues to investigate complex industrial systems. However, the relationship between the forced and damped behaviour and nonsmooth modes is not fully understood yet.