Farklı Uç Koşullarına sahip Uç Kütleli bir Ankastre Kirişin Eğilme Titreşimleri için Geliştirilmiş Yeni Bir Yöntem

Yıl 2025, ERKEN GÖRÜNÜM, 1 - 1

Orçun Biçer , Mustafa Yıldız , Sadettin Orhan

https://doi.org/10.2339/politeknik.1726887

Öz

Tek serbestlik dereceli toplu model, uç kütlesi olan konsol kirişin eğilme titreşimini incelerken basitliği nedeniyle sıklıkla kullanılır. Bu amaçla, bazı varsayımlarla eşdeğer tek serbestlik dereceli bir model elde edilir. Bu varsayımlar sonucun doğruluğunu etkiler. Bu çalışma, uç kütlesinin kiriş kütlesine oranının bir fonksiyonu olarak tanımlanan ve ilk doğal frekans değerini elde etmek için kullanılan orantı parametresi η için yeni bir formülasyon önermektedir. Önerilen formüle bağlı olarak, ilk doğal frekansın kütle oranına göre hesaplanması basitleştirilmiş ve sonuçlar iyileştirilmiştir. Orantı parametresinin yeni formülasyonu, etkinliğini doğrulamak için literatürde incelenen referans modele uygulanmıştır. Doğrulandıktan sonra, üç yeni karmaşık durum ele alınmış ve karşılık gelen eşdeğer yay sabiti ve doğal frekans parametresi elde edilmiştir. Son olarak, yalnızca yükseklik konikliği, yalnızca genişlik konikliği ve çift koniklikten oluşan uç kütlesi olan ve olmayan üç farklı konik durum için ilk boyutsuz doğal frekanslar elde edilmiştir. Sonuçların literatürde sunulan teorik frekanslara yakın olduğu görülebilir.

Anahtar Kelimeler

Uç Kütleli Ankastre Kiriş , Doğal Frekans , Kütle-Yay Sistemi , Eğilme Titreşimleri , Ayrık Titreşim Modeli

An Improved Approach for Bending Vibrations of Cantilever Beam with Tip Mass at Different End Conditions

Öz

Single dof lumped model is frequently utilized due to its simlicity, when studying bending vibration of cantilever beam with a tip mass. For this purpose, an equivalent single dof model is obtained with some assumptions. These assumptions effect the accuracy of the result. This study proposes a new formulation of proportion parameter η which is defined as a function of ratio of the tip mass to the beam mass and used to obtain the value of the first natural frequency. Depending on proposed formula, calculation of the first natural frequency with respect to the mass ratio was simplified and the results were improved. The new formulation of the proportion parameter was applied to the reference model studied in literature to verify its effectiveness. After it is verified, three complex new cases were considered and the corresponding equivalent spring constant and natural frequency parameter were obtained. Lastly, the first non-dimensional natural frequencies were obtained for three different tapered cases with and without tip mass consisting of only height taper, only width taper and double taper. One can see that the results are close to theoretical frequencies presented in the literature.

Anahtar Kelimeler

Cantilever beam with tip mass , Natural frequency , Mass-spring system , Lateral Vibrations , Discrete vibration model

Kaynakça

• [1] Alvarez, M., & Lechuga, L. M. “Microcantilever-based platforms as biosensing tools.” Analyst, 135(5), 827-836 (2010).
• [2] Spletzer, M., Raman, A., Sumali, H., & Sullivan, J. P. “Highly sensitive mass detection and identification using vibration localization in coupled microcantilever arrays.” Applied Physics Letters, 92(11) (2008).
• [3] ] Koç M.A., Eroğlu M., Esen İ., “Dynamic analysis of high speed train moving on perforated Timoshenko and Euler–Bernoulli beams” International Journal of Mechanics and Materials in Design, 18 893-917, (2022).
• [4] Esen İ., Abdelrhmaan A. A., Eltaher M.A., “Free vibration and buckling stability of FG nanobeams exposed to magnetic and thermal fields” Engineering with Computers, 38, 3463-3482, (2022).
• [5] Esen İ, Koç M. A., Çay Y., “Finite element formulation and analysis of a functionally graded Timoshenko beam subjected to an accelerating mass including inertial effects of the mass” Latin American Journal of Solids and Structures, 15(10), 1-18, (2018).
• [6] Esen İ, Özarpa C., Eltaher M.A., “Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment” Composite Structures, 261, 1-15, (2021).
• [7] Koç M. A., Esen İ, Eroğlu M., Çay Y., Çerlek Ö., “Dynamic Analysis of Flexible Structures Under The Influence of Moving Multiple Vehicles” El-Cezerî Journal of Science and Engineering 5(1), 176-181, (2018).
• [8] Esen İ. “Dynamic Response of a Beam Due to an Accelerating Moving Mass Using Moving Finite Element Approximation” Mathematical and Computational Applications, 16(1), 171-182, (2011).
• [9] Kösedağ E., Ekici R., “Free Vibration Analysis of Foam-Core Sandwich Structures” Journal of Polytechnic, 24(1), 69-74, (2021).
• [10] Rao, Singiresu S. Mechanical vibrations. (2001).
• [11] William, T. Thomson. Theory of Vibration With Applications. PRENTICE-HALL, Incorporated, (1988).
• [12] Gürgöze, M. "On the eigenfrequencies of a cantilever beam with attached tip mass and a spring-mass system." Journal of Sound and Vibration 190(2) 149-162, (1996).
• [13] Gürgöze, M. "On the representation of a cantilevered beam carrying a tip mass by an equivalent spring–mass system." Journal of sound and vibration 282(1-2) 538-542, (2005).
• [14] ] Ercoli, L., and P. A. A. Laura. "Analytical and experimental investigation on continuous beams carrying elastically mounted masses." Journal of Sound and Vibration 114(3) 519-533, (1987).
• [15] Mabie, H. H., and C. B. Rogers. "Transverse vibrations of tapered cantilever beams with end loads." The Journal of the Acoustical Society of America 36.(3) 463-469, (1964).
• [16] Mabie, H. H., and C. B. Rogers. "Transverse vibrations of double‐tapered cantilever beams with end support and with end mass." The Journal of the Acoustical Society of America 55(5) 986-991, (1974).
• [17] Kim, Jae Eun. "On the equivalent mass-spring parameters and assumed mode of a cantilevered beam with a tip mass." Journal of Mechanical Science and Technology 31 1073-1078, (2017).