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1

Determining Frequency-Energy Dependencyof Nonlinear Normal Modes and Internal ResonancesUsing a Numerical Independent Approach

Ghariblou Arash, Mahmoudi Mussa

Computer Assisted Methods in Engineering and Science

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2022

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Vol. 29, no. 3

261--292

EN

Application of linear normal modes to the nonlinear area provides an in-depth investigation of structures. In this paper, a straightforward approach is proposed to investigatenonlinear normal modes (NNMs) thoroughly and focus on all possible solutions and bifurcations, independent of all initial assumptions and prior solutions. In this context, afterdiscretization of the response domain over an appropriate resolution, a periodicity algorithm is suggested to capture the solutions that meet the NNMs criteria. Afterward, thefrequency and energy of the system during accepted responses and degrees of freedom(DOFs)’ relations are derived. Finally, after verifying the proposed approach and acquir-ing new internal resonances, the frequency-energy plots and NNMs of a nonlinear elasticsystem with more substantial nonlinearities and a two-story steel structure with nonlinearmaterial are studied. It is worth noting that the periodicity algorithm and capturing allpossible solutions and bifurcations are among the apparent novelties of the current paper.

2

Influence of nonlinear foundation on Timoshenko beam vibration

Bar A., Bar O.

Mechanics / AGH University of Science and Technology

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2008

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Vol. 27, no. 3

96-103

EN

The goal of this paper is presentation of the physical properties of the thick beam on nonlinear foundation with infrequent analyzed free boundary condition. Galerkin's method with two-elements basis is applied. This paper justifies this choice too. Nonlinear effects such as internal resonance and discontinuous of amplitude frequency characteristic is observed. The proper choice of parameters ensures the ratio first to second natural frequencies equal to 3. This is one of the conditions satisfying existing of the internal resonance. In the amplitude-frequency characteristic near first resonance there appear breaks and their numbers grows with the increase of nonlinearity of the foundation.

PL

Celem pracy jest opis właściwości fizycznych krępej belki wspartej na nieliniowym podłożu, przy rzadko analizowanych warunkach brzegowych opisujących swobodne końce. Zastosowano metodę Galerkina opartą na dwuelementowej bazie. Praca zawiera uzasadnienie takiego wyboru. Zaobserwowano efekty nieliniowe także jak rezonans wewnętrzny oraz nieciągłości charakterystyki amplitudowo--częstotliwościowej. Odpowiedni dobór parametrów pozwala uzyskać stosunek pierwszej i drugiej częstości własnej rzędu 3, co jest jednym z warunków istnienia rezonansu wewnętrznego. Pokazano też, że na charakterystyce w pobliżu pierwszego rezonansu pojawiają się "przerwy", których liczba zwiększa się wraz ze wzrostem udziału nieliniowości w oddziaływaniu podłoża.

3

Three-to-one internal resonances in a curved beam resting on an elastic foundation

Oz H. R., Ozkaya E.

International Journal of Applied Mechanics and Engineering

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2005

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Vol. 10, no 4

667-678

EN

Transverse vibrations of curved beams are investigated. Different curvatures are considered. The beam is simply supported and resting on a nonlinear elastic foundation. The method of multiple scales is used in the analysis. A three-to-one internal resonance case is studied. It is possible when one of the mode numbers is three times the other mode. Amplitude and phase modulation equations are obtained. Steady state solutions and stability are discussed. A three-to-one internal resonance is possible for all types of curvatures under the assumptions in the article.