Mathematical Problems in Engineering / 2011 / Article / Tab 1

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 π‘₯ , 𝑋 π‘œ , π‘Œ π‘œ π‘₯ π‘₯ = 𝐿 𝑇 , 𝑋 π‘œ = 𝑋 π‘œ 𝐿 𝑇 , π‘Œ π‘œ = π‘Œ π‘œ / 𝐿 𝑇 ,
Time 𝑑 𝑑 𝜏 = 𝐿 𝑇 ξ‚™ 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 𝑒 , 𝑣 𝑒 = 𝑒 / 𝐿 𝑇 , 𝑣 = 𝑣 / 𝐿 𝑇
spatial derivatives of displacement 𝑒 ξ…ž , 𝑒 ξ…ž ξ…ž , 𝑣 ξ…ž , 𝑣 ξ…ž ξ…ž 𝑒 ξ…ž = πœ• 𝑒 πœ• π‘₯ = 𝑒 ξ…ž , 𝑒 ξ…ž ξ…ž = πœ• 2 𝑒 πœ• π‘₯ 2 = 𝐿 𝑇 𝑒 ξ…ž ξ…ž , 𝑣 ξ…ž = πœ• 𝑣 πœ• π‘₯ = 𝑣 ξ…ž , 𝑣 ξ…ž ξ…ž = πœ• 2 𝑣 πœ• π‘₯ 2 = 𝐿 𝑇 𝑣 ξ…ž ξ…ž
Time derivatives of displacement Μ‡ 𝑒 , ̈ 𝑒 , Μ‡ 𝑣 , ̈ 𝑣 Μ‡ β€Œ πœ• 𝑒 = 𝑒 ξ‚™ πœ• 𝜏 = Μ‡ 𝑒 𝜌 𝐸 , ̈ πœ• 𝑒 = 2 𝑒 πœ• 𝜏 2 = 𝐿 𝑇 𝜌 𝐸 Μ‡ ̈ 𝑒 , πœ• 𝑣 = 𝑣 = Μ‡ 𝑣 ξ‚™ πœ• 𝜏 𝜌 𝐸 , ̈ πœ• 𝑣 = 2 𝑣 πœ• 𝜏 2 = 𝐿 𝑇 𝜌 𝐸 ̈ 𝑣
Force and moment 𝑓 𝑖 𝑗 , π‘š 𝑗 𝑓 𝑖 𝑗 = 𝑓 𝑖 𝑗 𝐸 𝐴 , ( 𝑖 = 1 , 2 ; 𝑗 = 1 , 2 )
Angular velocity Ξ© π‘˜ = Ξ© 𝐿 𝑇 √ 𝜌 / 𝐸
Natural frequency πœ” 𝐾 = πœ” 𝐿 𝑇 √ 𝜌 / 𝐸