A linear complementarity problem formulation for periodic solutions to unilateral contact problems

Abstract

Presented is an approach for finding periodic responses of structural systems subject to unilateral contact conditions. No other non-linear terms, e.g. large displacements or strains, hyper-elasticity, plasticity, etc. are considered. The excitation period due to various forcing conditions-from harmonic external or contact forcing due to a moving contact interface-is discretized in time, such that the quantities of interest-displacement, velocity, acceleration as well as contact force-can be approximated through time-domain schemes such as backward difference, Galerkin, and Fourier. The solution is assumed to exist and is defined on a circle with circumference T to directly enforce its periodicity. The strategy for approximating time derivative terms within the discretized period, i.e. velocity and acceleration, is hence circulant in nature. This results in a global circulant algebraic system of equations with inequalities that can be translated into a unique linear complementarity problem (LCP). The LCP is then solved by dedicated and established methods such as Lemke's algorithm. This allows for the computation of approximate periodic solutions exactly satisfying unilateral contact constraints on a discrete time set. The implementation efficiency and accuracy are discussed in comparison to classical time marching techniques for initial value problems combined with a Lagrange multiplier contact treatment. The LCP algorithm is validated using a simple academic problem. The extension to large-scale systems is made possible through the implementation of a Craig-Bampton type modal component synthesis. The method shows applicability to industrial rotor/casing contact set-ups as shown by studying a compressor blade. A good agreement to time marching simulations is found, suggesting a viable alternative to time marching or Fourier-based harmonic balance simulations.

Development of a Harmonic Balance Method-Based Numerical Strategy for Blade-Tip/Casing Interactions: Application to NASA Rotor 37

“…The aforementioned developments are recalled for the sake of completeness as they have already been widely used in the literature [14] for various types of nonlinearities. Nonetheless, when considering the HBM with contacts nonlinearities, it has been shown that it is prone to the Gibbs phenomenon [13] and, as a result, performs poorly [7].…”

“…Secondly, conclusions drawn from published experimental studies put forward that witnessed interactions are often related to transient or diverging motions which may not be captured with frequency-domain approaches. Thirdly, while frequency-domain approaches are often viewed as more computationally efficient than time integration methods, they still represent a computational challenge when applied to complex mechanical systems with a large number of nonlinear degrees of freedom (dof), even more so when one of the two impacting structures is assumed to be perfectly rigid [7]. For all these reasons, most of the published numerical work that was compared to experimental observations relied on time integration methods.…”

“…Its use for the blade-tip/casing interface remains however very limited [12] due to the numerical severity of the associated contact problem. The sensitivity of the HBM to the well-known Gibbs phenomenon [13] has been underlined by many researchers as a roadblock for the application of this method to blade-tip/casing contacts [7].…”

Modeling and Nonlinear Response of the Cam-Follower Oblique-Impact System

“…A linear complementarity problem (LCP) is a problem of the form y = Mx + b, y ≥ 0, x ≥ 0, and y x = 0, where M is an × matrix and x, y, and b are an -dimensional vector [8]. LCP equations may have unique solution, no solution, or multiple solutions.…”

Nonsmooth contact dynamics for the numerical simulation of collisions in musical string instruments

“…Nonsmooth methods have been succesfully applied in a variety of contexts ranging from granular media (Renouf et al, 2004), geomaterials (Jean, 1995), multibody dynamics (Chen et al, 2013) to realistic simulations of hair motions and living systems (Acary et al, 2014;Bertails-Descoubes et al, 2011). In the field of vibration, the method has been applied to rotor/casing contacts in (Meingast et al, 2014) as well as to string vibrations in (Ahn, 2007), where the study was however limited to the case of a perfect string without stiffness and a frictionless contact. In the area of musical acoustics, the action of a grand piano has been recently simulated efficiently by using a nonsmooth approach (Thorin et al, 2017).…”