On the use of complex numbers in equations of nonlinear structural dynamics

Tatzko, Sebastian; Jahn, Martin

doi:10.1016/j.ymssp.2019.02.041

Cited by 9 publications

(4 citation statements)

References 11 publications

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“…If the mass of the wire sections between nozzles and the workpiece ( h 1 + h 2 ), which is varied in the range of 3–5 mm ( H’ n - H n ), and the mass losses on tool-electrode wear [143, 144, 145, 146] are neglected, then the amplitude of oscillation of the wire tool may be expressed as a complex amplitude of the harmonic signal [147, 148, 149, 150, 151]:Aˆn=A0·etrueβ¯τ,where A 0 is the amplitude in the direction of wire feed, which is limited by the discharge gap Δ and is less than Δ, A n ≤ A 0 for stable EDM, βtrue¯ is the damping coefficient expressed as a complex number in the form a + bi , and τ is the period of oscillation T .…”

Section: Resultsmentioning

confidence: 99%

Electrical discharge machining of ceramic nanocomposites: sublimation phenomena and adaptive control

Grigoriev

Kozochkin

Porvatov

et al. 2019

Heliyon

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The productivity of electrical discharge machining (EDM) is relatively low owing to the natural laws of electrical erosion. Precise EDM demands uninterrupted control of the discharge gap and adjustment of process parameters. It is particularly critical for processing large workpieces with complex linear surfaces and for materials with threshold conductivities such as the new advanced ceramic nanocomposites Al2O3+TiC and Al2O3+SiCw+TiC(30–40%). In these cases, adequate flushing of erosion products is hampered by the geometry of the working space or by the small value of the required discharge gap, which does not exceed 2.2–2.5 μm. The methods of adaptive control in modern computer numerical control systems of EDM equipment based on measuring the electrical parameters in the working zone have been shown to be ineffective in the cases described above. This study aims to investigate the natural phenomena of material sublimation under discharge pulses for conductive ceramics and nanocomposites. The measured conductivities of the samples are higher than the percolation threshold. However, the question of machinability remains open owing to detected processing interruptions and poor quality of machined surfaces. New knowledge on EDM of conductive ceramics and nanocomposites can improve the final quality of the machined surfaces and productivity of the method by the introduction of advanced monitoring and control methods based on acoustic emissions. The manuscript presents an up-to-date overview and current state of the research on the subject area. The obtained morphology of the samples and discussion of the findings complete the experimental part of the study. The scientific basis for a new type of adaptive control system is provided. This can improve the effectiveness of parameter control for machining conductive ceramics and nanocomposites and contribute to an increase in the EDM performance for the most critical cases.

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Section: Resultsmentioning

confidence: 99%

Electrical discharge machining of ceramic nanocomposites: sublimation phenomena and adaptive control

Grigoriev

Kozochkin

Porvatov

et al. 2019

Heliyon

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“…Here, the nonlinear Duffing term is approximated by its fundamental harmonic known as Harmonic Balance, cp. [21]. The reference amplitude can now be written as a function of the vibration amplitude in the form

{\hat{x}}_{ref} = \hat{x} \frac{\sqrt{{(k + K_{p} - Ω^{2} m + α \frac{3}{4} {\hat{x}}^{2})}^{2} + Ω^{2} {(d + K_{d})}^{2}}}{\sqrt{K_{p}^{2} + Ω^{2} K_{d}^{2}}} .

…”

Section: Feedback Control and Choice Of Controller Gainsmentioning

confidence: 99%

Continuation methods for lab experiments of nonlinear vibrations

2023

Self Cite

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In this work, we will give an overview of our recent progress in experimental continuation. First, three different approaches are explained and compared which can be found in scientific papers on the topic. We then show S‐Curve measurements of a Duffing oscillator experiment for which we derived optimal controller gains analytically. The derived formula for stabilizing PD‐controller gains makes trial and error search for suitable values unnecessary. Since feedback control introduces higher harmonics in the driving signal, we consider a harmonization of the forcing signal. This harmonization is important to reduce shaker‐structure interaction in the treatment of nonlinear frequency responses. Finally, the controlled nonlinear testing and harmonization is enhanced by a continuation algorithm adapted from numerical analysis and applied to a geometrically nonlinear beam test rig for which we measure the nonlinear forced response directly in the displacement‐frequency plane.

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“…The almost-period vibration is also analysed using HBM in [6][7][8]. The HBM is described in a number of papers [9][10][11][12] and monographs [13,14]. Due to the abundance of literature on the subject, only selected and most recent items are mentioned above.…”

Section: Introductionmentioning

confidence: 99%

“…The exponential version of HBM together with the continuation method was used in [11] to analyse periodic responses of rotor/stator dynamics. The discussed method was also utilized in papers [9,12] to discuss some problems of nonlinear dynamics. Using the exponential version of HBM, the steady state vibration is also analysed in [4].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

Lewandowski

2022

Meccanica

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This paper presents the application of an exponential version of the harmonic balance method to the analysis of steady state vibration of geometrically nonlinear systems. A detailed description of the method and of the corresponding numerical procedure is provided. The von Karman theory is used to describe the effects of geometric nonlinearity. The material of the beams is modelled with the help of the Zener model using the fractional calculus. The problem is solved using an exponential version of the harmonic balance method. In the above-mentioned version, the complex calculus is used in contrast to the ordinary harmonic balance method, where the steady state vibrations are described with the help of the trigonometric functions. It significantly simplifies derivation of the amplitude equations. Moreover, the exponential version of the harmonic balance method is more elegant in comparison with the ordinary one. A detailed derivation of the amplitude equations is presented. The modified continuation method is proposed to solve the nonlinear amplitude equations and to determine the response curves. Moreover, the results of the exemplary calculation are presented and compared with known results in order to justify the efficiency and the correctness of the proposed approach.

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On the use of complex numbers in equations of nonlinear structural dynamics

Cited by 9 publications

References 11 publications

Electrical discharge machining of ceramic nanocomposites: sublimation phenomena and adaptive control

Electrical discharge machining of ceramic nanocomposites: sublimation phenomena and adaptive control

Continuation methods for lab experiments of nonlinear vibrations

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

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