Augmenting the Harmonic Balance Method by Stability Analysis and Error Estimation and its Application to Vibro-Impact Processes
Final Report Abstract
In this project, the harmonic balance method was extended by an efficient computation of Urabe's error bound and a stability analysis using Chebychev polynomials. Unfortunately, it turned out that a rigorous error bound requires such a high harmonic order that it is of little use for many models used in current practice. This motivates future research into revolutionary approaches to model reduction. On the other hand, a number of representative numerical examples show that Chebychev-based stability analysis does indeed provide a significant speed-up. However, like harmonic balance itself, this method becomes inefficient when an extremely high harmonic order is required, such as in the case of vibro-impact processes, which, however, were in the focus of the present project. Due to the limitations of the extended harmonic balance with respect to vibro-impact systems, a method for numerical time-step integration was developed. The approach is based on the massless boundary concept and component mode synthesis. The results show that the developed method has excellent energy conservation properties and outstanding convergence behavior. Compared to known methods such as time-step integration with a massless boundary and harmonic balance, the computational effort is reduced by typically 1-2 orders of magnitude. The developed prediction method based on numerical time step integration was validated experimentally. For this purpose, a system of two cantilevered beams was considered, which have different but closely spaced natural frequencies and undergo frictional collisions at the free end. The predictions agree very well with the measurements. It is known that isolated regions of amplitude-frequency responses can occur near primary resonances in the case of strong interactions of internally resonant modes or nonlinear damping. A methodology has been developed to systematically analyze the emergence and disappearance of such isolated regions in an experiment. For this purpose, the backbone curve is tracked using amplitude and phase control. The data obtained shows which excitation levels lead to the formation of isolated regions or to merging with the main branch. Further analysis of the data allows to characterize possible internal resonances and amplitude-dependent damping. With the developed method, new findings on the formation of isolated regions were obtained for a test rig with two unilaterally clamped beams interacting via a unilateral spring.
Publications
-
A massless boundary component mode synthesis method for elastodynamic contact problems. Computers & Structures, 260, 106698.
Monjaraz, Tec C.D.; Gross, J. & Krack, M.
-
How Intrusive Are Accelerometers for Measuring Nonlinear Vibrations? A Case Study on a Compressor Blade Subjected to Vibro-Impact Dynamics. Journal of Vibration and Acoustics, 144(4).
Woiwode, Lukas; Müller, Florian; Groß, Johann; Scheel, Maren & Krack, Malte
-
Prediction and validation of the strongly modulated forced response of two beams undergoing frictional impacts. Mechanical Systems and Signal Processing, 180, 109410.
Monjaraz-Tec, C.; Kohlmann, L.; Schwarz, S.; Hartung, A.; Gross, J. & Krack, M.
-
Are Chebyshev-based stability analysis and Urabe’s error bound useful features for Harmonic Balance?. Mechanical Systems and Signal Processing, 194, 110265.
Woiwode, Lukas & Krack, Malte
-
Experimentally uncovering isolas via backbone tracking.. Journal of Structural Dynamics.
Woiwode, Lukas & Krack, Malte
