Abstract
Efficiently predicting the vibratory responses of flexible structures which experience unilateral contact is becoming of high engineering importance. An example of such a system is a rotor blade within a turbine engine; small operating clearances and varying loading conditions often result in contact between the blade and the casing. The method of weighted residuals is a effective approach to simulating such behaviour as it can efficiently enforce time-periodic solutions of lightly damped, flexible structures experiencing unilateral contact. The Harmonic Balance Method (HBM) based on Fourier expansion of the sought solution is a common formulation, though it is hypothesized wavelet bases that can sparsely define nonsmooth solutions may be superior. This is investigated herein using an axially vibrating rod with unilateral contact conditions. A distributional formulation in time is introduced allowing periodic, square-integrable trial functions to approximate the second-order equations. The mixed wavelet Petrov-Galerkin solutions are found to yield consistent or better results than HBM, with similar convergence rates and seemingly more accurate contact force prediction.
Highlights
Predicting the vibratory responses of flexible structures which experience unilateral contact is becoming of high engineering importance
The second technique is based on weighted residual formulations, which are of interest in the current investigation. This method involves approximating the solution using a set of time-dependent basis functions, called trial functions, and enforcing the respective residual error to be orthogonal to an independent set of weighting functions [6, 7]
For this number of basis functions Harmonic Balance Method (HBM) (Fourier:Fourier) approximates the tip displacements well at both 150 Hz and 1275 Hz compared to the time-stepping solution
Summary
Predicting the vibratory responses of flexible structures which experience unilateral contact is becoming of high engineering importance. Structural displacements and velocities which satisfy these contact conditions are known to respectively feature absolute continuity and bounded variation only [2] This implies displacements are not necessarily differentiable everywhere in the domain and velocities may exhibit jumps; these types of problems are generally referred to as nonsmooth [3, 4]. Unlike the shooting method which can become numerically sensitive to possible jumps in the velocity field, weighted residual techniques directly enforce the periodicity conditions while the remaining unilateral contact constraints and governing local equations of motion are satisfied in a weak integral sense [8]. The main goal of the current work is to explore relevant basis functions whose order of smoothness can be adapted to a particular system to attain accurate approximations and rapid convergence
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