Laboratory notes Report 2

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MECHANICAL SYSTEM DYNAMICS Finite Elements Method (FEM) in structural dynamics: software implementation in Matlabenvironment –part 1 I. La Paglia 1.Mesh generation [USER INPUT: *.inp] 2.Definition of the global and local reference systems [USER INPUT: *.inp] 3.Removal of external constraints and introduction of correspondingconstraint forces [USER INPUT: *.inp] 4.Energy functions formulation in the local nodal coordinates of each element [FEM PROGRAM: loadstructure()] 5.Coordinate transformation from the local to the global referencesystem [FEM PROGRAM: assem(), callingel_tra()] 6.Matrix assembling, for the entire structure [FEM PROGRAM: assem()] Procedure for building a FE model of a structure2 Main.m % structure datam = … % check max element lengthLmax= … % build *.inpfile (mesh, constraints) loadstructure()% nodes definition% elements definition % draw structuredis_stru() % build and assemble matriciesassem()el_tra() % build M and K local ref. % assemble total M and K I. La PagliaMechanical Systems Dynamics •Element length The element must work in quasi-static region! The element first natural frequency is 𝜔𝑘(1)should be sufficiently grater than the maximum frequency of interest Ω𝑚𝑎𝑥 Typically, we can produce a mesh such that 𝜔𝑘(1) Ω𝑚𝑎𝑥≥1.5 thus 3 Must consider •Discontinuities 𝑚1 𝑚2 Property changeGeometry change I. La PagliaMechanical Systems Dynamics Mesh generation 𝐿𝑚𝑎𝑥= 𝜋2 1.5Ω𝑚𝑎𝑥 𝐸𝐽 𝑚𝑚𝑖𝑛 Example -Properties of the system 4 A simply supported aluminum beam with rectangular constant cross-section I. La PagliaMechanical Systems Dynamics Ω𝑚𝑎𝑥=1002𝜋→𝐿𝑚𝑎𝑥= 𝜋2 1.5Ω𝑚𝑎𝑥 𝐸𝐽 𝑚𝑚𝑖𝑛 =348𝑚𝑚 5 We must define:▪Nodes:oConstraints(x, y, θ)oCoordinates(x, y)▪Elements:oConnected nodesoProperties(m, EA, EJ) 4 beamelements(300 mm) and 5 nodes *NODESn –constr.(x,y,theta) –coor. x,y11…