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MME 615 Handout 14 - FINITE ELEMENT METHOD Handout 14 MME...

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FINITE ELEMENT METHOD Handout 14 - MME 615: Advanced Vibration Kumar V. Singh - 1 - Introduction to Finite Element (Computational Framework: Energy Method) Powerful numerical technique that uses energy methods, calculus of variation and interpolation schemes in solving large boundary value problems in which a continuous structure is divided into well defined substructures (finite elements) having elemental material and geometrical parameters These elements are defined by nodes which connect the neighboring elements, possesses degree of freedom (translation and/or rotation) and at which the boundary conditions and loads are applied. Extremely useful for approximating the governing equations of complicated, large and complex geometry structures Provides a computational framework which can address wide range of static and dynamics problems Energy associated with the continuous and discrete structures Kinetic energy of the continuous bar or beam: 2 0 , 1 2 L bar beam u x t T T A x x dx t Potential energy of the continuum bar: 2 0 , 1 2 L bar u x t U E x A x dx x Potential energy of the continuum beam:     2 0 , 1 2 L beam w x t U E x I x dx x x Kinetic energy of the discrete mass: 2 1 2 mass i i T m u Potential energy of the discrete stiffness: 2 1 2 springs i i U k u Non-conservative work done due to external force , x t f : 0 , , L W x t u x t dx f Find the Total Kinetic energy of the system, potential energy of the system and non-conservative forces and apply Lagrange method: i i i i d T T U F dt u u u
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FINITE ELEMENT METHOD Handout 14 - MME 615: Advanced Vibration Kumar V. Singh - 2 - Main idea: Figure 8.1 [1] A structure is divided into small pieces (finite elements) with nodes which connects with neighboring elements and represents the motion of element. Describe the physical parameters of each element Assemble the elements at the nodes in forming a approximated discrete system representing whole structure Solve the system of equations involving unknown quantities at the nodes (e.g., displacements, slope, strains, stresses, natural frequencies, mode shapes etc.) FEM FORMULATION FOR AXIALLY VIBRATING BAR Consider an axially loaded bar as shown in the figure here. The Potential energy of the axially loaded bar corresponding to the exact solution u(x) can be obtained as, 2 0 1 2 L du U EA dx dx , And the work done due to distributed forces can be obtained as, 0 ( ) L W F x u(x)dx These quantities associated with the axially loaded bar can also be computed by choosing a suitable approximated solution w(x) as follows, 2 0 0 1 2 ( ) L L dw U w EA dx dx W w F x w(x)dx x L F ) ( , x A E y ) ( x u
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