In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales (a perturbation technique). The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and frequencyresponse curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.
The free vibration of stepped nanobeams embedded in the elastic foundation was investigated using Eringen's nonlocal elasticity theory. It is fixed at the system ends with a simple-simple support. The stepped nanbeam’s equations of motion are obtained by using Hamilton's principle. Multi-time scale, which is the perturbation methods, was used for the analytical solution of the equations. To observe the effects of nano size effect, elastic basis coefficient and step location, natural frequencies of the first three modes of the system were obtained for different non-local parameter values, elastic foundation coefficients, step rates and step positions. In the results, it was seen that the non-local parameter had a negative effect on the natural frequency. The elastic foundation coefficient has been shown to reduce vibration amplitudes.
Bu çalışmada, simetrik ve farklı oryantasyon açılarına sahip iki tabakalı kompozit kirişlerin dinamik analizi numerik olarak incelenmiştir. İlk kısımda gerçek boyutlara sahip tabakalı kirişin sonlu elemanlar metodu ile analitik çözümü yapılmıştır. Analitik çözümde kiriş Euler-Bernoulli kiriş teorisi kirişi kabul edilmiştir. İkinci kısımda ise sistem sönümsüz serbest titreşime maruz bırakılarak dinamik analizi yapılmıştır. Sistemin numerik analizi için matematik analiz programı olan MATLAB program dili kullanılmıştır. İki tabakalı çeşitli sınır şartlarına sahip kirişlerin; farklı açılarda ve uzunluk-kalınlık oranlarında doğal frekansları elde edilmiştir. Tabakalı kompozit kirişlerde büyük genlikli titreşimler oldukça etkilidir. Bu sebeple frekanslar ilk sekiz mod için tablo haline getirilip yorumlanmıştır.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.