Abstract
This study is devoted to the investigation of the vibration of a cracked cantilever beam under moving mass load. The present formulation contains inertial, centripetal and Coriolis forces that depend on mass and the velocity of the moving load. The existence of crack induces a local flexibility which is a function of the crack depth, thereby changing its vibration behavior and the eigen-values of the system. The response of the system is obtained in terms of Duhamel integral. The differential equation which involves complicated terms on the right side is solved via an iterative procedure. It has been shown that the centripetal and Coriolis forces make an effect to decrease the deformations on the beam since the deformed beam remains concave during the passage of the moving load. It has also been detected that the previous solutions for the case of moving constant force had several mistakes. The results are exemplified for various values of the variables. Santrauka Straipsnyje nagrinėjami įtrūkusios gembinės sijos, kurią veikia judančios masės apkrova, virpesiai. Šiuo atveju veikia inercinės, įcentrinės ir Koriolio jėgos, kurios priklauso nuo judančios apkrovos masės ir judėjimo greičio. Įtrūkis sukelia vietinį sijos lankstumą, kuris priklauso nuo įtrūkio gylio, tokiu būdu keičia virpesių pobūdį ir sistemos reikšmes. Sistemos atsakas nustatomas taikant Duhamelo integralą. Diferencialinė lygtis, kurios dešinėje pusėje yra sudėtingos sąlygos, sprendžiama taikant iteracijas. Nustatyta, kad įcentrinė ir Koriolio jėgos sumažina sijos deformacijas, nes deformuota sija išlieka išsigaubusi, kai per ją juda masės apkrova. Taip pat nustatyta, kad ankstesniuose sprendimuose judančios nekintamos masės apkrovų atvejais buvo keletas klaidų. Kaip pavyzdžiai pateikiami rezultatai esant skirtingoms kintamųjų reikšmėms.
Highlights
There has been much work on the cracked structures since the crack influence the static and dynamic response of the mechanical system
The occurrence of crack at the beam induces a local flexibility which is a function of the crack depth, thereby changing its dynamic behavior and the eigen-values of the system
While some of these researchers were dealt with the detection of crack (Rizos, Aspragathos 1990; Liang et al 1991; Chondros, Dimarogonas 1980), some others were based on investigating the effects of cracks on the frequencies of the beam (Dimarogonas 1996; Lin, Chang 2006; Shifrin, Ruotolo 1999)
Summary
There has been much work on the cracked structures since the crack influence the static and dynamic response of the mechanical system. Several techniques were proposed to determine the eigen-functions of the cracked structures While some of these researchers were dealt with the detection of crack (Rizos, Aspragathos 1990; Liang et al 1991; Chondros, Dimarogonas 1980), some others were based on investigating the effects of cracks on the frequencies of the beam (Dimarogonas 1996; Lin, Chang 2006; Shifrin, Ruotolo 1999). Mahmoud and Abou Zaid (2002) used an iterative modal analysis approach to determine the cracked beam’s response Most of these works have analyzed the problem numerically or hybrid numerically. Lin and Chang (2006) analyzed the forced response of a cracked cantilever beam under a concentrated moving load. An analytical approach is presented for investigating the dynamic response of cracked beams under moving mass load.
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