Page - 786 - in Book of Full Papers - Symposium Hydro Engineering

2. THEORETICAL EQUATION OF DEEP BEAM NATURAL FREQUENCY According to Pedroso [4], the vibration equation of a deep beam, with constant properties along its length, considering the flexural, shear deformation, and rotational inertia of the cross-section effects and am applied dynamic distributed load is given by Eq. [1]: 𝐸 𝐼 ∙ 𝜕 4𝑦 𝜕 𝑥 4 −(𝑞 − ?̅? ∙ 𝜕 2𝑦 𝜕 𝑡 2 ) − 𝐽 ∙ 𝜕 4𝑦 𝜕 𝑥 2𝜕 𝑡 2 + 𝐸 𝐼 𝛾 ∙ 𝜕 2 𝜕 𝑡 2 (𝑞 − ?̅? ∙ 𝜕 2𝑦 𝜕 𝑡 2 ) − 𝐽 𝛾 ∙ 𝜕 2 𝜕 𝑡 2 (𝑞 − ?̅? ∙ 𝜕 2𝑦 𝜕 𝑡 2 ) = 0 [1] Where ?̅? is the mass by unit length; 𝛾 = 𝑘 ′ ∙ 𝐺 ∙ 𝐴 , where 𝑘 ′ the shear correction factor of the section, 𝐴 is the section area and 𝐺 is the transverse elastic modulus of the material; and 𝑞 = 𝑓 (𝑥 ,𝑡 ) is the dynamic distributed load. It is useful to state the meaning of each term in the Eq. [1]. The first term (Eq. [2]) represents the basic flexural theory (Euler’s beam). The second term (Eq. [3]) represents the consideration of rotational inertia of the cross-section effect. The third term (Eq. [4]) represents the shear deformation effect. The fourth term (Eq. [5]), represents the coupling between the rotational inertia and shear effects. 𝐸 𝐼 ∙ 𝜕 4𝑦 𝜕 𝑥 4 −(𝑞 − ?̅? ∙ 𝜕 2𝑦 𝜕 𝑡 2 ) [2] 𝐽 ∙ 𝜕 4𝑦 𝜕 𝑥 2𝜕 𝑡 2 [3] 𝐸 𝐼 𝛾 ∙ 𝜕 2 𝜕 𝑡 2 (𝑞 − ?̅? ∙ 𝜕 2𝑦 𝜕 𝑡 2 ) [4] 𝐸 𝐼 𝛾 ∙ 𝜕 2 𝜕 𝑡 2 (𝑞 − ?̅? ∙ 𝜕 2𝑦 𝜕 𝑡 2 ) [5] To obtain the mode shape function of free vibration, one can apply the separation of variables procedure to decouple the solution of Eq. [1] into a function of x (mode shape) and t (time-dependent amplitude) [2]. Assuming a harmonic response for the time-dependent amplitude, the application of separation of variables into Eq. [1] written for free vibration, i.e. 𝑞 = 0, yields Eq. [6]: 𝜙 (𝑥 )𝐼 𝑉 −𝑎 4 ∙ 𝜙 (𝑥 ) + 𝑎 4 ∙ 𝑟 2 ∙ 𝜙 (𝑥 )𝐼 𝐼 + ?̅? ∙𝜔 2 𝛾 ∙ [𝑎 4 ∙ 𝑟 2 ∙ 𝜙 (𝑥 ) + 𝜙 (𝑥 )𝐼 𝐼 ] = 0 [6] Where 𝜙 (𝑥 ) is the mode shape; 𝜙 (𝑥 )𝑖 denotes the i-th derivative of 𝜙 (𝑥 )𝑖 relative to x; 𝑟 is the radius of gyration; and 𝑎 is a convenient constant defined in Eq. [7]: 𝑎 = ?̅? ∙𝜔 2 𝐸 𝐼 [7] For simplicity it is assumed a mode shape form of a simply supported beam, since analytical solution of Eq. [6] is complex for other support conditions. The 786