Computational Engineering: BEAMS -- 2
Budget: €30 – €250 EUR
(To read properly the info open the attached file Assignment 2 - Beams.zip)
Consider a beam with the cross section depicted in Fig. 1 rigidly attached on one end and subjected to different loads on the other end. The beam is made of an isotropic material with Young’s modulus ? = 69 GPa, Poisson’s ratio ? = 0.27 and density ? = 2000 kg/m3.
(Drawings on the PDF document attached)
The following is asked: (a) Obtain the properties of the cross-section. Note: consider shear and torsion correction factors of ?? = 2???⁄?, ?? = h?h⁄?, and ?? = 0.01. (b) Consider a 100-elements discretization of the beam and compute the resulting mass and stiffness matrices. For the static case, obtain the following plots: i. ? /? and? /? vs.?,asaresultofapplyingaload? . ???? ? ii. ? /? and ? /? vs. ?, as a result of applying a load ? . ???? ? iii. ??/?? vs. ?, as a result of applying a torque ??. (c) Plot the first 3 vibration modes for: i. Shear deflection in ?: ?? ii. Shear deflection in ?: ?? iii. Bending deflection and angle in ?: ?? , ?? iv. Bending deflection and angle in ?: ?? , ?? v. Torsion angle: ??
(d) Obtain the frequency response (for a relevant range of frequencies) resulting from solving the dynamic system for the following cases: i. ?tip/? and ?tip/? vs. ?, as a result of applying a load ? ????. ???? ? ii. ?tip/? and ?tip/? vs. ?, as a result of applying a load ? ????. ???? ? iii. ?tip/? vs. ?, as a result of applying a torque ? ????. ??? For each case, try to approximate the results using a reduced set of relevant modes through a modal projection of the system of equations.
Consider a beam with the cross section depicted in Fig. 1 rigidly attached on one end and subjected to different loads on the other end. The beam is made of an isotropic material with Young’s modulus ? = 69 GPa, Poisson’s ratio ? = 0.27 and density ? = 2000 kg/m3.
(Drawings on the PDF document attached)
The following is asked: (a) Obtain the properties of the cross-section. Note: consider shear and torsion correction factors of ?? = 2???⁄?, ?? = h?h⁄?, and ?? = 0.01. (b) Consider a 100-elements discretization of the beam and compute the resulting mass and stiffness matrices. For the static case, obtain the following plots: i. ? /? and? /? vs.?,asaresultofapplyingaload? . ???? ? ii. ? /? and ? /? vs. ?, as a result of applying a load ? . ???? ? iii. ??/?? vs. ?, as a result of applying a torque ??. (c) Plot the first 3 vibration modes for: i. Shear deflection in ?: ?? ii. Shear deflection in ?: ?? iii. Bending deflection and angle in ?: ?? , ?? iv. Bending deflection and angle in ?: ?? , ?? v. Torsion angle: ??
(d) Obtain the frequency response (for a relevant range of frequencies) resulting from solving the dynamic system for the following cases: i. ?tip/? and ?tip/? vs. ?, as a result of applying a load ? ????. ???? ? ii. ?tip/? and ?tip/? vs. ?, as a result of applying a load ? ????. ???? ? iii. ?tip/? vs. ?, as a result of applying a torque ? ????. ??? For each case, try to approximate the results using a reduced set of relevant modes through a modal projection of the system of equations.