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1

The 1/3rd subharmonic and 3rd superharmonic resonance of a shape memory alloy (SMA) laminated beam

Zhang Xia Hui, Gao Ming, Hao Ying

Journal of Theoretical and Applied Mechanics

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2021

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Vol. 59 nr 1

27--41

EN

This paper examines the 1/3rd subharmonic resonance and the 3rd superharmonic resonance of simply-supported shape memory alloy (SMA) laminated beams. First, the dynamic equation for SMA laminated beams under transverse load is established using physical equations, force equilibrium conditions, the compatibility equation of deformation, and a constitutive model of SMA polynomial functions. Then, a differential equation for transverse vibration of the SMA laminated beams is derived by the Galerkin process assuming the boundary conditions for simply-supported beams. Next, the amplitude-frequency response equations for the 1/3rd subharmonic resonance and the 3rd superharmonic resonance of these beams are derived by an averaging method before their respective transition sets are calculated, and their amplitude-frequency response diagrams were plotted using singularity theory. The results show two different types of amplitude-frequency responses to nonlinear vibration under the 1/3rd subharmonic resonance and the 3rd superharmonic resonance: quasi-linear and hard characteristic. In the quasi-linear area, SMA thickness A does not make much difference to the response of the system, whereas in the hard-characteristics area, under the same excitation amplitude f, the resonance frequency increases with A. In the nonlinear area, SMA can obviously reduce vibration amplitude, but it is not obvious for the 1/3rd subharmonic resonance. The nonlinear solution of both the 1/3rd subharmonic resonance and 3rd superharmonic resonance are stable.

2

Analysis of the primary and superharmonic contact resonances – Part 1.

Kostek R.

Journal of Theoretical and Applied Mechanics

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2013

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Vol. 51 nr 2

475--486

EN

This paper presents results of numerical and analytical investigations of non-linear normal contact microvibrations excited by a harmonic force in a system of two bodies in planar contact. The system models, for example, the slide unit of machine tools or positioning systems. The main aim of the computational analysis is to present the evolution of the resonance phenomena under various amplitudes of the excitation force. The studies show that, beside the primary resonance, a number of superharmonic resonances appear, which take place in the single-degree-of-freedom non-linear system excited by a harmonic force. Thus, in a resonance plot, a number of peaks is observed. The superharmonic (ultraharmonic) resonances take place at excitation frequencies being below the natural frequency, and becoming stronger with the increase of the excitation amplitude. The resonances are coupled with complex non-linear phenomena like: asymmetry of vibrations, bending resonance peak, bi-stability, multi-stability and loss of contact, which are presented and described in this paper.

PL

W artykule tym przedstawiono wyniki badań numerycznych oraz analitycznych drgań kontaktowych normalnych. Drgania te zostały wzbudzone przez silę harmoniczną w układzie składającym się z dwóch ciał, które tworzą parę cierną. Głównym celem pracy było zbadanie zmian rezonansów kontaktowych pod wpływem wzrostu amplitudy wymuszenia. Oprócz rezonansu głównego zaobserwowano także rezonanse ultraharmoniczne, które są wzbudzane dla częstotliwości będących poniżej częstotliwości własnej układu. Amplituda rezonansów ultraharmonicznych staje się stopniowo większa wraz ze wzrostem amplitudy wymuszenia. Ponadto zaobserwowano asymetrię drgań kontaktowych, odrywanie się ciał od siebie, wiele atraktorów dla takiego samego wzbudzenia oraz zaginanie się pików rezonansowych. Wymienione zjawiska zaobserwowano dla nieliniowego układu o jednym stopniu swobody.

3

An analysis of the primary and superharmonic contact resonances – Part 2

Kostek R.

Journal of Theoretical and Applied Mechanics

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2013

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Vol. 51 nr 3

687--696

EN

This paper presents results of investigations of non-linear normal contact microvibrations excited by a harmonic force in a system of two bodies in planar contact. The system models, for instance, slide units of machine tools or their positioning systems. The main aim of the computational analysis is to present resonance graphs and time histories obtained with numerical and perturbation methods. Good agreement between the perturbation and numerical results leads to the conclusion that the perturbation solution is correct. The obtained perturbation solution describes well both the primary resonance and the superharmonic resonances. Characteristic phenomena typical for non-linear vibrations are depicted, viz. asymmetry of vibrations, multi-harmonic vibrations, non-elliptical phase portraits, loss of contacts, bending resonance peaks, bi-stabilities, and multi-stability.

4

An analysis of the primary and superharmonic contact resonances – Part 3

Kostek R.

Journal of Theoretical and Applied Mechanics

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2013

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Vol. 51 nr 4

799--812

EN

This paper presents a study of non-linear normal contact vibrations excited by an external harmonic force in a system containing two bodies being in planar contact. The system models, for instance, the slide unit of machine tools or positioning systems. The presented results, which are obtained both with numerical and perturbation methods, show clearly the evolution of resonance phenomena under various excitation amplitudes. Apart from the primary resonance, a number of super-harmonic resonances has been excited in the nonlinear single-degree-of-freedomsystem. Hence, a resonance graph contains a number of peaks being below the natural frequency. The contact vibrations are associated with strongly nonlinear phenomena like: asymmetry of vibrations, loss of contact, bending resonance peak, multi-stability, period-doubling bifurcations, chaotic vibrations, which are far from linear dynamics. These phenomena are presented and described in this article.