Mathematical Problems in Engineering

Research Article

A Corotational Finite Element Method Combined with Floating Frame Method for Large Steady-State Deformation and Free Vibration Analysis of a Rotating-Inclined Beam

Table 1

Dimensionless variables.
VariablesDimensionless variables
Coordinates 𝑥 , 𝑋 𝑜 , 𝑌 𝑜 𝑥 𝑥 = 𝐿 𝑇 , 𝑋 𝑜 = 𝑋 𝑜 𝐿 𝑇 , 𝑌 𝑜 = 𝑌 𝑜 / 𝐿 𝑇 ,
Time 𝑡 𝑡 𝜏 = 𝐿 𝑇 𝐸 𝜌
Length of beam element 𝐿 𝐿 = 𝐿 / 𝐿 𝑇
Area moment of inertia 𝐼 𝐼 𝐼 = 𝐴 𝐿 2 𝑇
Radius of hub 𝑅 𝑅 = 𝑅 / 𝐿 𝑇
Displacements 𝑢 , 𝑣 𝑢 = 𝑢 / 𝐿 𝑇 , 𝑣 = 𝑣 / 𝐿 𝑇
spatial derivatives of displacement 𝑢 , 𝑢 , 𝑣 , 𝑣 𝑢 = 𝜕 𝑢 𝜕 𝑥 = 𝑢 , 𝑢 = 𝜕 2 𝑢 𝜕 𝑥 2 = 𝐿 𝑇 𝑢 , 𝑣 = 𝜕 𝑣 𝜕 𝑥 = 𝑣 , 𝑣 = 𝜕 2 𝑣 𝜕 𝑥 2 = 𝐿 𝑇 𝑣
Time derivatives of displacement ̇ 𝑢 , ̈ 𝑢 , ̇ 𝑣 , ̈ 𝑣 ̇ ‌ 𝜕 𝑢 = 𝑢 𝜕 𝜏 = ̇ 𝑢 𝜌 𝐸 , ̈ 𝜕 𝑢 = 2 𝑢 𝜕 𝜏 2 = 𝐿 𝑇 𝜌 𝐸 ̇ ̈ 𝑢 , 𝜕 𝑣 = 𝑣 = ̇ 𝑣 𝜕 𝜏 𝜌 𝐸 , ̈ 𝜕 𝑣 = 2 𝑣 𝜕 𝜏 2 = 𝐿 𝑇 𝜌 𝐸 ̈ 𝑣
Force and moment 𝑓 𝑖 𝑗 , 𝑚 𝑗 𝑓 𝑖 𝑗 = 𝑓 𝑖 𝑗 𝐸 𝐴 , ( 𝑖 = 1 , 2 ; 𝑗 = 1 , 2 )
Angular velocity Ω 𝑘 = Ω 𝐿 𝑇 √ 𝜌 / 𝐸
Natural frequency 𝜔 𝐾 = 𝜔 𝐿 𝑇 √ 𝜌 / 𝐸