Anders Thorin

Nonsmooth Modal Analysis of Piecewise-Linear Impact Oscillators

Periodic solutions of autonomous and conservative second-order dynamical systems of finite dimension

Nonsmooth modal analysis: Investigation of a 2-dof spring-mass system subject to an elastic impact law

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Nonlinear modal analysis of a one-dimensional bar undergoing unilateral contact via the time-domain boundary element method

undergoing one unilateral contact condition are investigated in continuous time. The unilateral constraint is complemented with a purely elastic impact law which preserves total energy. The dynamics is linear when there is no contact. The number

The Wave Finite Element Method applied to a one-dimensional linear elastodynamic problem with unilateral constraints

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Nonsmooth Modal Analysis: From the Discrete to the Continuous Settings

of impacts per period arises as a natural parameter of the proposed formulation. Interestingly, the existence of the targeted periodic solutions is essentially governed by a system of only k-1 nonlinear equations with

Thermomechanical model reduction for efficient simulations of rotor-stator contact interaction

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Nonsmooth Thermoelastic Simulations of Blade-Casing Contact Interactions

unknowns, regardless of the number of degrees of freedom. This serves to prove that the phase space is populated by one-dimensional continua of periodic solutions generating invariant manifolds which can be understood as nonsmooth modes of vibration in the context of vibration analysis. Additionally, these equations provide an efficient and systematic way of calculating nonsmooth modes of vibration. They also demonstrate the existence of i...

Nonsmooth modal analysis of an elastic bar subject to a unilateral contact constraint

The well-known concept of normal mode for linear systems has been extended to the framework of nonlinear dynamics over the course of the 20th century, initially by Lyapunov, and later by Rosenberg and a growing community of researchers in modal and vibration analysis. This effort has mainly targeted nonlinear smooth systems — the velocity is continuous and differentiable in time — even though systems presenting nonsmooth occurrences have been increasingly studied in the last decades to face the growing industrial need of unilateral contact and friction simulations. Yet, these systems have nearly never been explored from the standpoint of modal analysis.This contribution addresses the notion of modal analysis of nonsmooth systems. Developments are illustrated on a seemingly simple 2-dof autonomous system, subject to unilateral constraints reflected by a perfectly elastic impact law. Even though friction is ignored, its dynamics appears to be extremely rich. Periodic solutions are sought for given numbers of impacts per period and nonsmooth modes are illustrated for one and two impacts per period in the form of two-dimensional manifolds in the phase space. Also, an unexpected bridge between these modes in the frequency-energy plots is observed.Copyright

Non-smooth dynamics for an efficient simulation of the grand piano action

Finite elements in space with time-stepping numerical schemes, even though versatile, face theoretical and numerical difficulties when dealing with unilateral contact conditions. In most cases, an impact law has to be introduced to ensure the uniqueness of the solution: total energy is either not preserved or spurious high-frequency oscillations arise. In this work, the Time Domain Boundary Element Method (TD-BEM) is shown to overcome these issues on a one-dimensional system undergoing a unilateral Signorini contact condition. Unilateral contact is implemented by switching between free boundary conditions (open gap) and fixed boundary conditions (closed gap). The solution method does not numerically dissipate energy unlike the Finite Element Method and properly captures wave fronts, allowing for the search of periodic solutions. Indeed, TD-BEM relies on fundamental solutions which are travelling Heaviside functions in the considered one-dimensional setting. The proposed formulation is capable of capturing main, subharmonic as well as internal resonance backbone curves useful to the vibration analyst. For the system of interest, the nonlinear modeshapes are piecewise-linear unseparated functions of space and time, as opposed to the linear mode shapes that are separated half sine waves in space and full sine waves in time.

Spectrum of an impact oscillator via nonsmooth modal analysis

The Wave Finite Element Method (WFEM) is implemented to accurately capture travelling waves propagating at a finite speed within a bouncing rod system and induced by unilateral contact collisions with a rigid foundation; friction is not accounted for. As opposed to the traditional Finite Element Method (FEM) within a time-stepping framework, potential discontinuous deformation, stress and velocity wave fronts are accurately described, which is critical for the problem of interest.A one-dimensional benchmark with an analytical solution is investigated. The WFEM is compared to two time-stepping solution methods formulated on a FEM semi-discretization in space: (1) an explicit technique involving Lagrange multipliers and (2) a non-smooth approach implemented in the Siconos package. Attention is paid to the Gibb’s phenomenon generated during and after contact occurrences together with the time evolution of the total energy of the system.It is numerically found that the WFEM outperforms the FEM and Siconos solution methods because it does not induce any spurious oscillations or dispersion and diffusion of the shock wave. Furthermore, energy is not dissipated over time. More importantly, the WFEM does not require any impact law to close the system of governing equations.Copyright