Nonlinear dynamics of mechanical systems with friction contacts: Coupled static and dynamic Multi-Harmonic Balance Method and multiple solutions

Abstract

Real applications in structural mechanics where the dynamic behaviour is linear are rare. Usually, structures are made of components assembled together by means of joints whose behaviour maybe highly nonlinear. Depending on the amount of excitation, joints can dramatically change the dynamic behaviour of the whole system, and the modelling of this type of constraint is therefore crucial for a correct prediction of the amount of vibration. The solution of the nonlinear equilibrium equations by means of the Harmonic Balance Method (HBM)is widely accepted as an effective approach to calculate the steady-state forced response in the frequency domain, in spite of Direct Time Integration (DTI). The state-of-the-art contact element used to model the friction forces at the joint interfaces is a node-to-node contact element, where the local contact compliance is modelled by means of linear springs and Coulomb's law is used to govern the friction phenomena. In the literature, when the HBM is applied to vibrating systems with joint interfaces and the state-of-the-art contact model is used, an uncoupled approach is mostly employed: the static governing equations are solved in advance to compute the pre-stress effects and then the dynamic governing equations are solved to predict the vibration amplitude of the system. As a result, the HBM steady-state solution may lead to a poor correlation with the DTI solution, where static and dynamic loads are accounted for simultaneously..In this paper, the HBM performances are investigated by comparing the uncoupled approach to a fully coupled static/dynamic approach. In order to highlight the main differences between the two approaches, a lumped parameter system, characterized by a single friction contact, is considered in order to show the different levels of accuracy that the proposed approaches can provide for different configurations.

Thermomechanical Component Mode Synthesis for Blade Casing Interaction Prediction

“…central finite differences [21,3]. Harmonic methods are used to model perioddic steady-state phenomena and have been applied to study blade-root / disk contacts [22] and friction phenomena in general [23] .…”

Nonlinear Forced Response of a Stator Vane With Multiple Friction Contacts Using a Coupled Static/Dynamic Approach

“…In order to reduce the calculation time of a numerical integration, the harmonic balance method (HBM) can be used to solve the equations of motion of the system [5,12,13,14]. This is possible due to the periodicity of the external excitation that involves also the periodicity of the displacements Q and the non-linear forces F nl .…”

“…The Equations (2)- (4) are replaced into the equations of motion (1) obtaining two sets of algebraic equations, a set of static algebraic equations and a set of algebraic complex dynamic equations [14]:…”

“…the overall balance conditions (static and dynamic) are computed together. This procedure will be referred as the 'coupled' calculation of the forced response, where the term coupled means that the static and the dynamic quantities involved in the dynamic response of the vibrating system are updated at each iteration and may change for different excitation frequencies [14]. In order to separate the solution of the non-linear part of the system from its linear part it is convenient to rearrange the balance equations (5) and (6): other, because the number of harmonics that compose the nonlinear contact forces depends on all the harmonics components of the displacements of the nonlinear dofs Q nl (t).…”

Comparison of two algorithms for Harmonic Balance and path continuation

“…Consequently, HB greatly gained importance in the field of (nonlinear) mechanical systems. For example, vibration problems of jointed structures with frictional contact interactions were thoroughly analyzed with HB, e. g. [5][6][7][8][9]. Besides, there are various review articles addressing the application of HB to a particular realm, e. g. on rotor dynamics [10] and brake squeal [11].…”