Rastgele ideal olmayan mesnet koşullarına sahip nanokirişlerin serbest titreşiminde belirsizlik analizi

Abstract

Bu çalışma, modifiye çift gerilme teorisi çerçevesinde, belirsiz ideal olmayan sınır koşullarına sahip nanokirişlerin serbest titreşim davranışını incelemektedir. Nanoölçekli mesnetlerde imalat kaynaklı değişkenliği temsil etmek amacıyla, ideal olmayan sınır parametreleri sınırlı Beta dağılımlı rassal değişkenler olarak modellenmiştir. Bu parametrelerin ortalama değerleri mesnet konfigürasyonuna göre belirlenirken, belirsizlik düzeyini kontrol etmek için yoğunlaşma parametresi kullanılmıştır. Çalışmanın temel özgünlüğü, modifiye çift gerilme teorisine dayalı boyut-bağımlı nanokiriş formülasyonunda ideal olmayan mesnet parametrelerinin stokastik olarak modellenmesidir. Ortaya çıkan özdeğer problemi, analitik karakteristik denklem ile Monte Carlo simülasyonunun birlikte kullanılmasıyla çözülmüştür. Her bir mesnet konfigürasyonu için 4000 Monte Carlo örneği üretilmiş ve ilk üç doğal frekans; olasılık yoğunluk fonksiyonları, kutu grafikleri ve %5-%95 güven aralığı kullanılarak istatistiksel olarak değerlendirilmiştir. Sonuçlar, sınır koşulu belirsizliğinin hem doğal frekans düzeylerini hem de bunların istatistiksel yayılımını önemli ölçüde etkilediğini göstermektedir. Ankastre-ankastre konfigürasyon en yüksek frekansları ve en geniş belirsizlik aralığını sergilerken; basit mesnetli konfigürasyon en düşük frekansları ve en dar yayılımı üretmektedir. Önerilen formülasyonun, nano ölçekli rezonatör, sensör ve aktüatör uygulamalarında mesnet belirsizliğinin etkilerinin değerlendirilmesine katkı sağlaması beklenmektedir.

Keywords

• Nanokirişler
• Modifiye çift gerilme teorisi
• İdeal olmayan sınır koşulları
• Sınır şartı belirsizliği
• Beta dağılımı
• Serbest titreşim

Uncertainty analysis of the free vibration of nanobeams with random non-ideal supports

Abstract

This study investigates the free vibration behavior of nanobeams with uncertain non-ideal boundary conditions within the framework of the modified couple stress theory. To represent manufacturing induced variability in nanoscale supports, the non-ideal boundary parameters are modeled as bounded Beta-distributed random variables. Their mean values are assigned according to the support configuration, while the concentration parameter controls the uncertainty level. The main novelty of this study is the stochastic modeling of non-ideal support parameters in a size-dependent nanobeam formulation based on the modified couple stress theory. The resulting eigenvalue problem is solved by combining the analytical characteristic equation with Monte Carlo simulation. For each support configuration, 4000 Monte Carlo samples are generated, and the first three natural frequencies are statistically evaluated using probability density functions, box plots, and 5th-95th percentile bounds. The results show that boundary condition uncertainty significantly affects both the natural frequency levels and their statistical spread. The clamped-clamped configuration exhibits the highest frequencies and widest uncertainty range, whereas the simply supported configuration produces the lowest frequencies and narrowest spread. The proposed formulation may support the evaluation of support uncertainty effects in nanoscale resonator, sensor, and actuator applications.

Keywords

• Nanobeams
• Modified couple stress theory
• Non-ideal boundary conditions
• Boundary condition uncertainty
• Beta distribution
• Free vibration

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