Free Vibration Analysis Of Axially Moving Functionally Graded Material Beams Under Thermal Shock And Scale Effect

Axially moving system and nano-structure are widely used in construction,machinery,aerospace and other engineering fields and it is important to analyze the vibration behavior of structure and system accurately and efficiently.In this paper,differential quadrature method(DQM)is used to solve the vibration behaviors of axially moving functionally gradient material(FGM)beams under thermal shock and scale effect.The main work includes:Based on the Timoshenko beam theory,free vibrations of axially moving functionally graded material beams under thermal shock were investigated for three different conditions with clamped-clamped,clamped-simply supported and simply supported-simply supported boundary.Using the Hamilton principle,the governing differential equation of free vibration of an axially moving functionally gradient material beam under thermal shock was derived and the one-dimensional heat conduction equation was solved by variable separation method.The beam was discretized along the length direction by using differential quadrature method and the original equation was turned into fourth order generalized characteristic value problem,then the dimensionless natural frequencies of free vibration of the functionally gradient material beam were solved and analyzed.The influences of the different heat flow input,graded index and dimensionless axially moving speed on natural frequencies of free vibration of the functionally gradient material beam were presented.The results show that the larger the thermal shock load is,the more obvious the effect is to reduce the natural frequency of the functionally gradient material beam.Under the condition of certain axially moving speeds and heat flow input,increasing gradient indexes of functionally graded material gradually will reduce the dimensionless natural frequencies of the beams.The functionally gradient material beam will slow down the thermal shock in a short period of time and the first order instability will take long time relative to homogeneous material.Under thermal shock,the functionally gradient material beam easier reach an instability state in axially moving.Based on Timoshenko beam theory and Eringen non-local theory,free vibrations of axially moving functionally graded material beam under scale effect were investigated for different boundary conditions.Considering that the performance of functionally graded beam varies exponentially along the thickness direction,based on Eringen non-local theory,using the Hamilton principle,the governing differential equation of free vibration of an axially moving functionally graded material beam under scale effect was derived.The beam was discretized along the length direction by using differential quadrature method,the natural frequencies of free vibration of the functionally graded material beam under scale effect were solved and characteristic analyzed,the results was compared with the existing literatures.The influences of the dimensionless axially moving speed,scale parameter and graded index on natural frequencies of free vibration of the functionally graded material beam were presented.The results show that under the condition of certain axially moving speeds and scale parameter,increasing material graded index gradually will reduce the natural frequencies of the beams,increasing scale parameter and axially moving speeds will reduce dimensionless natural frequencies,and the fastest reduction of dimensionless natural frequencies under the conditions with clamped-clamped.