Equal[iseR]

Introduction

Equal[iseR] is a frequency-domain solver for non-smooth systems of ODE/DAE targetting the vibration analysis of structural mechanical systems subject to unilateral and frictional contact occurrences. It is an adaptation of the well-known Harmonic Balance Method. It relies on [Equal]ity-based versions of the governing equations solved in a weighted [Resi]dual sense.

Unilateral contact and friction commonly take the form of differential inclusions which can be recast into more conventional equality-based identities through the use of non-smooth projections. Such equations are then approximately transformed using classical weighted residual techniques. The resulting system of nonsmooth/nonlinear equations is solved using the classical Hybrid Powell solver.

Supporting paper

The paper A compact, equality-based weighted residual formulation for periodic solutions of systems undergoing frictional occurrences (see also the version available on HAL: hal-04189699) introduces the methodology and attentant equations.

History

2024.03.27 first public release (version 1.0)

Version 1.0

The project comes with two versions of the solution procedure

• Matlab Three versions are available (Tested on Matlab R2021b):
  • ImplicitEuler Time-marching solution method based on the first-order Implicit Euler scheme for comparison purposes with the frequency-domain solution strategies.
  • WeightedResiduals Frequency-domain solution strategy based on the proposed weighted-residual formulation. Integrals are computed using classical (recursive) quadrature schemes and the nonlinear equations are solved using the Hybrid Powell (or dog-leg) nonlinear solver. Used in the initial version of the above paper.
  • FFT Same as WeightedResiduals but the integrals are computed using the FFT algorithm. This version of the solver is the fastest.
• Python One version is available (Tested on Python 3.8 and 3.11):
  • FFT Essentially the same as the Matlab FFT even though the iFFT operation involved in the algorithm is computed differently. Plotting features limited to the friction force but mapping to the displacements and velocities is straightforward (see Matlab scripts).

Authors

They are listed in the licence.